First formula, in copyable Mathematica format:

TraditionalForm@ HoldForm[ Sum[-(((-1)^ n (a^2 d - 3 a^3 d + 2 a^4 d + 4 a^3 b d - 4 a^4 b d - 4 a^2 b^2 d + 4 a^3 b^2 d - a c d + 2 a^2 c d + a^3 c d - 2 a^4 c d - 4 a^2 b c d + 4 a^4 b c d + 4 a b^2 c d - 4 a^3 b^2 c d + a c^2 d - 3 a^2 c^2 d + 2 a^3 c^2 d + 4 a^2 b c^2 d - 4 a^3 b c^2 d - 4 a b^2 c^2 d + 4 a^2 b^2 c^2 d - a d^2 + 5 a^2 d^2 - 4 a^3 d^2 - 8 a^2 b d^2 + 8 a^3 b d^2 + 4 a b^2 d^2 - 4 a^2 b^2 d^2 - a c d^2 - 3 a^2 c d^2 + 4 a^3 c d^2 + 4 a b c d^2 + 4 a^2 b c d^2 - 8 a^3 b c d^2 - 4 a b^2 c d^2 + 4 a^2 b^2 c d^2 + 2 a c^2 d^2 - 2 a^2 c^2 d^2 - 4 a b c^2 d^2 + 4 a^2 b c^2 d^2 - 2 a d^3 + 2 a^2 d^3 + 4 a b d^3 - 4 a^2 b d^3 + 2 a c d^3 - 2 a^2 c d^3 - 4 a b c d^3 + 4 a^2 b c d^3 + 5 a^2 e - 25 a^3 e + 44 a^4 e - 32 a^5 e + 8 a^6 e + 16 a^2 b e - 52 a^3 b e + 52 a^4 b e - 16 a^5 b e + 12 a^2 b^2 e - 20 a^3 b^2 e + 8 a^4 b^2 e - 5 a c e + 32 a^2 c e - 69 a^3 c e + 58 a^4 c e - 16 a^5 c e - 16 a b c e + 68 a^2 b c e - 80 a^3 b c e + 28 a^4 b c e - 12 a b^2 c e + 24 a^2 b^2 c e - 12 a^3 b^2 c e - 7 a c^2 e + 25 a^2 c^2 e - 26 a^3 c^2 e + 8 a^4 c^2 e - 16 a b c^2 e + 28 a^2 b c^2 e - 12 a^3 b c^2 e - 4 a b^2 c^2 e + 4 a^2 b^2 c^2 e + 10 a^2 d e - 54 a^3 d e + 72 a^4 d e - 24 a^5 d e - 16 a b d e + 104 a^2 b d e - 152 a^3 b d e + 48 a^4 b d e - 32 a b^2 d e + 72 a^2 b^2 d e - 24 a^3 b^2 d e + 2 c d e - 28 a c d e + 116 a^2 c d e - 148 a^3 c d e + 48 a^4 c d e + 16 b c d e - 112 a b c d e + 168 a^2 b c d e - 40 a^3 b c d e + 24 b^2 c d e - 48 a b^2 c d e + 10 c^2 d e - 52 a c^2 d e + 72 a^2 c^2 d e - 24 a^3 c^2 d e + 16 b c^2 d e - 32 a b c^2 d e - 8 b^2 c^2 d e + 16 a b^2 c^2 d e - 4 d^2 e + 22 a d^2 e - 6 a^2 d^2 e - 48 a^3 d^2 e + 24 a^4 d^2 e - 68 a b d^2 e + 148 a^2 b d^2 e - 48 a^3 b d^2 e + 16 b^2 d^2 e - 48 a b^2 d^2 e + 16 a^2 b^2 d^2 e + 8 c d^2 e - 70 a c d^2 e + 130 a^2 c d^2 e - 48 a^3 c d^2 e + 48 b c d^2 e - 92 a b c d^2 e - 4 a^2 b c d^2 e + 16 a b^2 c d^2 e + 16 c^2 d^2 e - 40 a c^2 d^2 e + 16 a^2 c^2 d^2 e + 16 a b c^2 d^2 e - 8 d^3 e + 16 a d^3 e + 8 a^2 d^3 e - 8 a^3 d^3 e + 16 b d^3 e - 48 a b d^3 e + 16 a^2 b d^3 e + 16 c d^3 e - 40 a c d^3 e + 16 a^2 c d^3 e + 16 a b c d^3 e + 25 a e^2 - 183 a^2 e^2 + 426 a^3 e^2 - 392 a^4 e^2 + 120 a^5 e^2 + 80 a b e^2 - 384 a^2 b e^2 + 528 a^3 b e^2 - 208 a^4 b e^2 + 60 a b^2 e^2 - 164 a^2 b^2 e^2 + 88 a^3 b^2 e^2 - 14 c e^2 + 165 a c e^2 - 521 a^2 c e^2 + 592 a^3 c e^2 - 208 a^4 c e^2 - 48 b c e^2 + 364 a b c e^2 - 652 a^2 b c e^2 + 304 a^3 b c e^2 - 40 b^2 c e^2 + 148 a b^2 c e^2 - 100 a^2 b^2 c e^2 - 22 c^2 e^2 + 138 a c^2 e^2 - 214 a^2 c^2 e^2 + 88 a^3 c^2 e^2 - 48 b c^2 e^2 + 164 a b c^2 e^2 - 100 a^2 b c^2 e^2 - 8 b^2 c^2 e^2 + 16 a b^2 c^2 e^2 - 4 d e^2 + 90 a d e^2 - 434 a^2 d e^2 + 672 a^3 d e^2 - 288 a^4 d e^2 - 32 b d e^2 + 412 a b d e^2 - 956 a^2 b d e^2 + 480 a^3 b d e^2 - 48 b^2 d e^2 + 288 a b^2 d e^2 - 192 a^2 b^2 d e^2 - 76 c d e^2 + 534 a c d e^2 - 1006 a^2 c d e^2 + 480 a^3 c d e^2 - 256 b c d e^2 + 868 a b c d e^2 - 484 a^2 b c d e^2 - 144 b^2 c d e^2 + 96 a b^2 c d e^2 - 116 c^2 d e^2 + 336 a c^2 d e^2 - 192 a^2 c^2 d e^2 - 128 b c^2 d e^2 + 96 a b c^2 d e^2 + 16 b^2 c^2 d e^2 + 32 d^2 e^2 - 8 a d^2 e^2 - 280 a^2 d^2 e^2 + 216 a^3 d^2 e^2 - 80 b d^2 e^2 + 528 a b d^2 e^2 - 368 a^2 b d^2 e^2 - 96 b^2 d^2 e^2 + 96 a b^2 d^2 e^2 - 128 c d^2 e^2 + 536 a c d^2 e^2 - 368 a^2 c d^2 e^2 - 288 b c d^2 e^2 + 208 a b c d^2 e^2 - 96 c^2 d^2 e^2 + 96 a c^2 d^2 e^2 + 48 d^3 e^2 - 48 a^2 d^3 e^2 - 96 b d^3 e^2 + 96 a b d^3 e^2 - 96 c d^3 e^2 + 96 a c d^3 e^2 + 32 e^3 - 446 a e^3 + 1542 a^2 e^3 - 1904 a^3 e^3 + 744 a^4 e^3 + 96 b e^3 - 924 a b e^3 + 1964 a^2 b e^3 - 1072 a^3 b e^3 + 64 b^2 e^3 - 432 a b^2 e^3 + 368 a^2 b^2 e^3 + 204 c e^3 - 1282 a c e^3 + 2222 a^2 c e^3 - 1072 a^3 c e^3 + 464 b c e^3 - 1732 a b c e^3 + 1252 a^2 b c e^3 + 240 b^2 c e^3 - 304 a b^2 c e^3 + 188 c^2 e^3 - 584 a c^2 e^3 + 368 a^2 c^2 e^3 + 256 b c^2 e^3 - 304 a b c^2 e^3 + 16 b^2 c^2 e^3 + 120 d e^3 - 1040 a d e^3 + 2328 a^2 d e^3 - 1368 a^3 d e^3 + 432 b d e^3 - 2128 a b d e^3 + 1776 a^2 b d e^3 + 320 b^2 d e^3 - 512 a b^2 d e^3 + 608 c d e^3 - 2328 a c d e^3 + 1776 a^2 c d e^3 + 1088 b c d e^3 - 1424 a b c d e^3 + 192 b^2 c d e^3 + 416 c^2 d e^3 - 512 a c^2 d e^3 + 192 b c^2 d e^3 - 16 d^2 e^3 - 512 a d^2 e^3 + 688 a^2 d^2 e^3 + 480 b d^2 e^3 - 864 a b d^2 e^3 + 128 b^2 d^2 e^3 + 544 c d^2 e^3 - 864 a c d^2 e^3 + 384 b c d^2 e^3 + 128 c^2 d^2 e^3 - 64 d^3 e^3 - 64 a d^3 e^3 + 128 b d^3 e^3 + 128 c d^3 e^3 - 368 e^4 + 2504 a e^4 - 4616 a^2 e^4 + 2440 a^3 e^4 - 752 b e^4 + 3184 a b e^4 - 2704 a^2 b e^4 - 352 b^2 e^4 + 672 a b^2 e^4 - 1040 c e^4 + 3624 a c e^4 - 2704 a^2 c e^4 - 1440 b c e^4 + 2224 a b c e^4 - 320 b^2 c e^4 - 512 c^2 e^4 + 672 a c^2 e^4 - 320 b c^2 e^4 - 832 d e^4 + 3584 a d e^4 - 3216 a^2 d e^4 - 1568 b d e^4 + 2848 a b d e^4 - 448 b^2 d e^4 - 1760 c d e^4 + 2848 a c d e^4 - 1280 b c d e^4 - 448 c^2 d e^4 - 320 d^2 e^4 + 960 a d^2 e^4 - 640 b d^2 e^4 - 640 c d^2 e^4 + 1536 e^5 - 5632 a e^5 + 4496 a^2 e^5 + 1952 b e^5 - 3360 a b e^5 + 448 b^2 e^5 + 2208 c e^5 - 3360 a c e^5 + 1408 b c e^5 + 448 c^2 e^5 + 2112 d e^5 - 3776 a d e^5 + 1664 b d e^5 + 1664 c d e^5 + 512 d^2 e^5 - 2752 e^6 + 4416 a e^6 - 1664 b e^6 - 1664 c e^6 - 1792 d e^6 + 1792 e^7 + 5 a^2 n - 25 a^3 n + 44 a^4 n - 32 a^5 n + 8 a^6 n + 16 a^2 b n - 52 a^3 b n + 52 a^4 b n - 16 a^5 b n + 12 a^2 b^2 n - 20 a^3 b^2 n + 8 a^4 b^2 n - 5 a c n + 32 a^2 c n - 69 a^3 c n + 58 a^4 c n - 16 a^5 c n - 16 a b c n + 68 a^2 b c n - 80 a^3 b c n + 28 a^4 b c n - 12 a b^2 c n + 24 a^2 b^2 c n - 12 a^3 b^2 c n - 7 a c^2 n + 25 a^2 c^2 n - 26 a^3 c^2 n + 8 a^4 c^2 n - 16 a b c^2 n + 28 a^2 b c^2 n - 12 a^3 b c^2 n - 4 a b^2 c^2 n + 4 a^2 b^2 c^2 n + 10 a^2 d n - 54 a^3 d n + 72 a^4 d n - 24 a^5 d n - 16 a b d n + 104 a^2 b d n - 152 a^3 b d n + 48 a^4 b d n - 32 a b^2 d n + 72 a^2 b^2 d n - 24 a^3 b^2 d n + 2 c d n - 28 a c d n + 116 a^2 c d n - 148 a^3 c d n + 48 a^4 c d n + 16 b c d n - 112 a b c d n + 168 a^2 b c d n - 40 a^3 b c d n + 24 b^2 c d n - 48 a b^2 c d n + 10 c^2 d n - 52 a c^2 d n + 72 a^2 c^2 d n - 24 a^3 c^2 d n + 16 b c^2 d n - 32 a b c^2 d n - 8 b^2 c^2 d n + 16 a b^2 c^2 d n - 4 d^2 n + 22 a d^2 n - 6 a^2 d^2 n - 48 a^3 d^2 n + 24 a^4 d^2 n - 68 a b d^2 n + 148 a^2 b d^2 n - 48 a^3 b d^2 n + 16 b^2 d^2 n - 48 a b^2 d^2 n + 16 a^2 b^2 d^2 n + 8 c d^2 n - 70 a c d^2 n + 130 a^2 c d^2 n - 48 a^3 c d^2 n + 48 b c d^2 n - 92 a b c d^2 n - 4 a^2 b c d^2 n + 16 a b^2 c d^2 n + 16 c^2 d^2 n - 40 a c^2 d^2 n + 16 a^2 c^2 d^2 n + 16 a b c^2 d^2 n - 8 d^3 n + 16 a d^3 n + 8 a^2 d^3 n - 8 a^3 d^3 n + 16 b d^3 n - 48 a b d^3 n + 16 a^2 b d^3 n + 16 c d^3 n - 40 a c d^3 n + 16 a^2 c d^3 n + 16 a b c d^3 n + 50 a e n - 366 a^2 e n + 852 a^3 e n - 784 a^4 e n + 240 a^5 e n + 160 a b e n - 768 a^2 b e n + 1056 a^3 b e n - 416 a^4 b e n + 120 a b^2 e n - 328 a^2 b^2 e n + 176 a^3 b^2 e n - 28 c e n + 330 a c e n - 1042 a^2 c e n + 1184 a^3 c e n - 416 a^4 c e n - 96 b c e n + 728 a b c e n - 1304 a^2 b c e n + 608 a^3 b c e n - 80 b^2 c e n + 296 a b^2 c e n - 200 a^2 b^2 c e n - 44 c^2 e n + 276 a c^2 e n - 428 a^2 c^2 e n + 176 a^3 c^2 e n - 96 b c^2 e n + 328 a b c^2 e n - 200 a^2 b c^2 e n - 16 b^2 c^2 e n + 32 a b^2 c^2 e n - 8 d e n + 180 a d e n - 868 a^2 d e n + 1344 a^3 d e n - 576 a^4 d e n - 64 b d e n + 824 a b d e n - 1912 a^2 b d e n + 960 a^3 b d e n - 96 b^2 d e n + 576 a b^2 d e n - 384 a^2 b^2 d e n - 152 c d e n + 1068 a c d e n - 2012 a^2 c d e n + 960 a^3 c d e n - 512 b c d e n + 1736 a b c d e n - 968 a^2 b c d e n - 288 b^2 c d e n + 192 a b^2 c d e n - 232 c^2 d e n + 672 a c^2 d e n - 384 a^2 c^2 d e n - 256 b c^2 d e n + 192 a b c^2 d e n + 32 b^2 c^2 d e n + 64 d^2 e n - 16 a d^2 e n - 560 a^2 d^2 e n + 432 a^3 d^2 e n - 160 b d^2 e n + 1056 a b d^2 e n - 736 a^2 b d^2 e n - 192 b^2 d^2 e n + 192 a b^2 d^2 e n - 256 c d^2 e n + 1072 a c d^2 e n - 736 a^2 c d^2 e n - 576 b c d^2 e n + 416 a b c d^2 e n - 192 c^2 d^2 e n + 192 a c^2 d^2 e n + 96 d^3 e n - 96 a^2 d^3 e n - 192 b d^3 e n + 192 a b d^3 e n - 192 c d^3 e n + 192 a c d^3 e n + 96 e^2 n - 1338 a e^2 n + 4626 a^2 e^2 n - 5712 a^3 e^2 n + 2232 a^4 e^2 n + 288 b e^2 n - 2772 a b e^2 n + 5892 a^2 b e^2 n - 3216 a^3 b e^2 n + 192 b^2 e^2 n - 1296 a b^2 e^2 n + 1104 a^2 b^2 e^2 n + 612 c e^2 n - 3846 a c e^2 n + 6666 a^2 c e^2 n - 3216 a^3 c e^2 n + 1392 b c e^2 n - 5196 a b c e^2 n + 3756 a^2 b c e^2 n + 720 b^2 c e^2 n - 912 a b^2 c e^2 n + 564 c^2 e^2 n - 1752 a c^2 e^2 n + 1104 a^2 c^2 e^2 n + 768 b c^2 e^2 n - 912 a b c^2 e^2 n + 48 b^2 c^2 e^2 n + 360 d e^2 n - 3120 a d e^2 n + 6984 a^2 d e^2 n - 4104 a^3 d e^2 n + 1296 b d e^2 n - 6384 a b d e^2 n + 5328 a^2 b d e^2 n + 960 b^2 d e^2 n - 1536 a b^2 d e^2 n + 1824 c d e^2 n - 6984 a c d e^2 n + 5328 a^2 c d e^2 n + 3264 b c d e^2 n - 4272 a b c d e^2 n + 576 b^2 c d e^2 n + 1248 c^2 d e^2 n - 1536 a c^2 d e^2 n + 576 b c^2 d e^2 n - 48 d^2 e^2 n - 1536 a d^2 e^2 n + 2064 a^2 d^2 e^2 n + 1440 b d^2 e^2 n - 2592 a b d^2 e^2 n + 384 b^2 d^2 e^2 n + 1632 c d^2 e^2 n - 2592 a c d^2 e^2 n + 1152 b c d^2 e^2 n + 384 c^2 d^2 e^2 n - 192 d^3 e^2 n - 192 a d^3 e^2 n + 384 b d^3 e^2 n + 384 c d^3 e^2 n - 1472 e^3 n + 10016 a e^3 n - 18464 a^2 e^3 n + 9760 a^3 e^3 n - 3008 b e^3 n + 12736 a b e^3 n - 10816 a^2 b e^3 n - 1408 b^2 e^3 n + 2688 a b^2 e^3 n - 4160 c e^3 n + 14496 a c e^3 n - 10816 a^2 c e^3 n - 5760 b c e^3 n + 8896 a b c e^3 n - 1280 b^2 c e^3 n - 2048 c^2 e^3 n + 2688 a c^2 e^3 n - 1280 b c^2 e^3 n - 3328 d e^3 n + 14336 a d e^3 n - 12864 a^2 d e^3 n - 6272 b d e^3 n + 11392 a b d e^3 n - 1792 b^2 d e^3 n - 7040 c d e^3 n + 11392 a c d e^3 n - 5120 b c d e^3 n - 1792 c^2 d e^3 n - 1280 d^2 e^3 n + 3840 a d^2 e^3 n - 2560 b d^2 e^3 n - 2560 c d^2 e^3 n + 7680 e^4 n - 28160 a e^4 n + 22480 a^2 e^4 n + 9760 b e^4 n - 16800 a b e^4 n + 2240 b^2 e^4 n + 11040 c e^4 n - 16800 a c e^4 n + 7040 b c e^4 n + 2240 c^2 e^4 n + 10560 d e^4 n - 18880 a d e^4 n + 8320 b d e^4 n + 8320 c d e^4 n + 2560 d^2 e^4 n - 16512 e^5 n + 26496 a e^5 n - 9984 b e^5 n - 9984 c e^5 n - 10752 d e^5 n + 12544 e^6 n + 25 a n^2 - 183 a^2 n^2 + 426 a^3 n^2 - 392 a^4 n^2 + 120 a^5 n^2 + 80 a b n^2 - 384 a^2 b n^2 + 528 a^3 b n^2 - 208 a^4 b n^2 + 60 a b^2 n^2 - 164 a^2 b^2 n^2 + 88 a^3 b^2 n^2 - 14 c n^2 + 165 a c n^2 - 521 a^2 c n^2 + 592 a^3 c n^2 - 208 a^4 c n^2 - 48 b c n^2 + 364 a b c n^2 - 652 a^2 b c n^2 + 304 a^3 b c n^2 - 40 b^2 c n^2 + 148 a b^2 c n^2 - 100 a^2 b^2 c n^2 - 22 c^2 n^2 + 138 a c^2 n^2 - 214 a^2 c^2 n^2 + 88 a^3 c^2 n^2 - 48 b c^2 n^2 + 164 a b c^2 n^2 - 100 a^2 b c^2 n^2 - 8 b^2 c^2 n^2 + 16 a b^2 c^2 n^2 - 4 d n^2 + 90 a d n^2 - 434 a^2 d n^2 + 672 a^3 d n^2 - 288 a^4 d n^2 - 32 b d n^2 + 412 a b d n^2 - 956 a^2 b d n^2 + 480 a^3 b d n^2 - 48 b^2 d n^2 + 288 a b^2 d n^2 - 192 a^2 b^2 d n^2 - 76 c d n^2 + 534 a c d n^2 - 1006 a^2 c d n^2 + 480 a^3 c d n^2 - 256 b c d n^2 + 868 a b c d n^2 - 484 a^2 b c d n^2 - 144 b^2 c d n^2 + 96 a b^2 c d n^2 - 116 c^2 d n^2 + 336 a c^2 d n^2 - 192 a^2 c^2 d n^2 - 128 b c^2 d n^2 + 96 a b c^2 d n^2 + 16 b^2 c^2 d n^2 + 32 d^2 n^2 - 8 a d^2 n^2 - 280 a^2 d^2 n^2 + 216 a^3 d^2 n^2 - 80 b d^2 n^2 + 528 a b d^2 n^2 - 368 a^2 b d^2 n^2 - 96 b^2 d^2 n^2 + 96 a b^2 d^2 n^2 - 128 c d^2 n^2 + 536 a c d^2 n^2 - 368 a^2 c d^2 n^2 - 288 b c d^2 n^2 + 208 a b c d^2 n^2 - 96 c^2 d^2 n^2 + 96 a c^2 d^2 n^2 + 48 d^3 n^2 - 48 a^2 d^3 n^2 - 96 b d^3 n^2 + 96 a b d^3 n^2 - 96 c d^3 n^2 + 96 a c d^3 n^2 + 96 e n^2 - 1338 a e n^2 + 4626 a^2 e n^2 - 5712 a^3 e n^2 + 2232 a^4 e n^2 + 288 b e n^2 - 2772 a b e n^2 + 5892 a^2 b e n^2 - 3216 a^3 b e n^2 + 192 b^2 e n^2 - 1296 a b^2 e n^2 + 1104 a^2 b^2 e n^2 + 612 c e n^2 - 3846 a c e n^2 + 6666 a^2 c e n^2 - 3216 a^3 c e n^2 + 1392 b c e n^2 - 5196 a b c e n^2 + 3756 a^2 b c e n^2 + 720 b^2 c e n^2 - 912 a b^2 c e n^2 + 564 c^2 e n^2 - 1752 a c^2 e n^2 + 1104 a^2 c^2 e n^2 + 768 b c^2 e n^2 - 912 a b c^2 e n^2 + 48 b^2 c^2 e n^2 + 360 d e n^2 - 3120 a d e n^2 + 6984 a^2 d e n^2 - 4104 a^3 d e n^2 + 1296 b d e n^2 - 6384 a b d e n^2 + 5328 a^2 b d e n^2 + 960 b^2 d e n^2 - 1536 a b^2 d e n^2 + 1824 c d e n^2 - 6984 a c d e n^2 + 5328 a^2 c d e n^2 + 3264 b c d e n^2 - 4272 a b c d e n^2 + 576 b^2 c d e n^2 + 1248 c^2 d e n^2 - 1536 a c^2 d e n^2 + 576 b c^2 d e n^2 - 48 d^2 e n^2 - 1536 a d^2 e n^2 + 2064 a^2 d^2 e n^2 + 1440 b d^2 e n^2 - 2592 a b d^2 e n^2 + 384 b^2 d^2 e n^2 + 1632 c d^2 e n^2 - 2592 a c d^2 e n^2 + 1152 b c d^2 e n^2 + 384 c^2 d^2 e n^2 - 192 d^3 e n^2 - 192 a d^3 e n^2 + 384 b d^3 e n^2 + 384 c d^3 e n^2 - 2208 e^2 n^2 + 15024 a e^2 n^2 - 27696 a^2 e^2 n^2 + 14640 a^3 e^2 n^2 - 4512 b e^2 n^2 + 19104 a b e^2 n^2 - 16224 a^2 b e^2 n^2 - 2112 b^2 e^2 n^2 + 4032 a b^2 e^2 n^2 - 6240 c e^2 n^2 + 21744 a c e^2 n^2 - 16224 a^2 c e^2 n^2 - 8640 b c e^2 n^2 + 13344 a b c e^2 n^2 - 1920 b^2 c e^2 n^2 - 3072 c^2 e^2 n^2 + 4032 a c^2 e^2 n^2 - 1920 b c^2 e^2 n^2 - 4992 d e^2 n^2 + 21504 a d e^2 n^2 - 19296 a^2 d e^2 n^2 - 9408 b d e^2 n^2 + 17088 a b d e^2 n^2 - 2688 b^2 d e^2 n^2 - 10560 c d e^2 n^2 + 17088 a c d e^2 n^2 - 7680 b c d e^2 n^2 - 2688 c^2 d e^2 n^2 - 1920 d^2 e^2 n^2 + 5760 a d^2 e^2 n^2 - 3840 b d^2 e^2 n^2 - 3840 c d^2 e^2 n^2 + 15360 e^3 n^2 - 56320 a e^3 n^2 + 44960 a^2 e^3 n^2 + 19520 b e^3 n^2 - 33600 a b e^3 n^2 + 4480 b^2 e^3 n^2 + 22080 c e^3 n^2 - 33600 a c e^3 n^2 + 14080 b c e^3 n^2 + 4480 c^2 e^3 n^2 + 21120 d e^3 n^2 - 37760 a d e^3 n^2 + 16640 b d e^3 n^2 + 16640 c d e^3 n^2 + 5120 d^2 e^3 n^2 - 41280 e^4 n^2 + 66240 a e^4 n^2 - 24960 b e^4 n^2 - 24960 c e^4 n^2 - 26880 d e^4 n^2 + 37632 e^5 n^2 + 32 n^3 - 446 a n^3 + 1542 a^2 n^3 - 1904 a^3 n^3 + 744 a^4 n^3 + 96 b n^3 - 924 a b n^3 + 1964 a^2 b n^3 - 1072 a^3 b n^3 + 64 b^2 n^3 - 432 a b^2 n^3 + 368 a^2 b^2 n^3 + 204 c n^3 - 1282 a c n^3 + 2222 a^2 c n^3 - 1072 a^3 c n^3 + 464 b c n^3 - 1732 a b c n^3 + 1252 a^2 b c n^3 + 240 b^2 c n^3 - 304 a b^2 c n^3 + 188 c^2 n^3 - 584 a c^2 n^3 + 368 a^2 c^2 n^3 + 256 b c^2 n^3 - 304 a b c^2 n^3 + 16 b^2 c^2 n^3 + 120 d n^3 - 1040 a d n^3 + 2328 a^2 d n^3 - 1368 a^3 d n^3 + 432 b d n^3 - 2128 a b d n^3 + 1776 a^2 b d n^3 + 320 b^2 d n^3 - 512 a b^2 d n^3 + 608 c d n^3 - 2328 a c d n^3 + 1776 a^2 c d n^3 + 1088 b c d n^3 - 1424 a b c d n^3 + 192 b^2 c d n^3 + 416 c^2 d n^3 - 512 a c^2 d n^3 + 192 b c^2 d n^3 - 16 d^2 n^3 - 512 a d^2 n^3 + 688 a^2 d^2 n^3 + 480 b d^2 n^3 - 864 a b d^2 n^3 + 128 b^2 d^2 n^3 + 544 c d^2 n^3 - 864 a c d^2 n^3 + 384 b c d^2 n^3 + 128 c^2 d^2 n^3 - 64 d^3 n^3 - 64 a d^3 n^3 + 128 b d^3 n^3 + 128 c d^3 n^3 - 1472 e n^3 + 10016 a e n^3 - 18464 a^2 e n^3 + 9760 a^3 e n^3 - 3008 b e n^3 + 12736 a b e n^3 - 10816 a^2 b e n^3 - 1408 b^2 e n^3 + 2688 a b^2 e n^3 - 4160 c e n^3 + 14496 a c e n^3 - 10816 a^2 c e n^3 - 5760 b c e n^3 + 8896 a b c e n^3 - 1280 b^2 c e n^3 - 2048 c^2 e n^3 + 2688 a c^2 e n^3 - 1280 b c^2 e n^3 - 3328 d e n^3 + 14336 a d e n^3 - 12864 a^2 d e n^3 - 6272 b d e n^3 + 11392 a b d e n^3 - 1792 b^2 d e n^3 - 7040 c d e n^3 + 11392 a c d e n^3 - 5120 b c d e n^3 - 1792 c^2 d e n^3 - 1280 d^2 e n^3 + 3840 a d^2 e n^3 - 2560 b d^2 e n^3 - 2560 c d^2 e n^3 + 15360 e^2 n^3 - 56320 a e^2 n^3 + 44960 a^2 e^2 n^3 + 19520 b e^2 n^3 - 33600 a b e^2 n^3 + 4480 b^2 e^2 n^3 + 22080 c e^2 n^3 - 33600 a c e^2 n^3 + 14080 b c e^2 n^3 + 4480 c^2 e^2 n^3 + 21120 d e^2 n^3 - 37760 a d e^2 n^3 + 16640 b d e^2 n^3 + 16640 c d e^2 n^3 + 5120 d^2 e^2 n^3 - 55040 e^3 n^3 + 88320 a e^3 n^3 - 33280 b e^3 n^3 - 33280 c e^3 n^3 - 35840 d e^3 n^3 + 62720 e^4 n^3 - 368 n^4 + 2504 a n^4 - 4616 a^2 n^4 + 2440 a^3 n^4 - 752 b n^4 + 3184 a b n^4 - 2704 a^2 b n^4 - 352 b^2 n^4 + 672 a b^2 n^4 - 1040 c n^4 + 3624 a c n^4 - 2704 a^2 c n^4 - 1440 b c n^4 + 2224 a b c n^4 - 320 b^2 c n^4 - 512 c^2 n^4 + 672 a c^2 n^4 - 320 b c^2 n^4 - 832 d n^4 + 3584 a d n^4 - 3216 a^2 d n^4 - 1568 b d n^4 + 2848 a b d n^4 - 448 b^2 d n^4 - 1760 c d n^4 + 2848 a c d n^4 - 1280 b c d n^4 - 448 c^2 d n^4 - 320 d^2 n^4 + 960 a d^2 n^4 - 640 b d^2 n^4 - 640 c d^2 n^4 + 7680 e n^4 - 28160 a e n^4 + 22480 a^2 e n^4 + 9760 b e n^4 - 16800 a b e n^4 + 2240 b^2 e n^4 + 11040 c e n^4 - 16800 a c e n^4 + 7040 b c e n^4 + 2240 c^2 e n^4 + 10560 d e n^4 - 18880 a d e n^4 + 8320 b d e n^4 + 8320 c d e n^4 + 2560 d^2 e n^4 - 41280 e^2 n^4 + 66240 a e^2 n^4 - 24960 b e^2 n^4 - 24960 c e^2 n^4 - 26880 d e^2 n^4 + 62720 e^3 n^4 + 1536 n^5 - 5632 a n^5 + 4496 a^2 n^5 + 1952 b n^5 - 3360 a b n^5 + 448 b^2 n^5 + 2208 c n^5 - 3360 a c n^5 + 1408 b c n^5 + 448 c^2 n^5 + 2112 d n^5 - 3776 a d n^5 + 1664 b d n^5 + 1664 c d n^5 + 512 d^2 n^5 - 16512 e n^5 + 26496 a e n^5 - 9984 b e n^5 - 9984 c e n^5 - 10752 d e n^5 + 37632 e^2 n^5 - 2752 n^6 + 4416 a n^6 - 1664 b n^6 - 1664 c n^6 - 1792 d n^6 + 12544 e n^6 + 1792 n^7) Pochhammer[1 - d + e, n] Pochhammer[ 1/2 + a - b - d + e, n] Pochhammer[1 + a - c - d + e, n] Pochhammer[1 + d + e, n] Pochhammer[1 + a + 2 e, 2 n] Pochhammer[1/2 + b + 2 e, 2 n] Pochhammer[1 + c + 2 e, 2 n])/((-d + e + n) (a - c - d + e + n) (d + e + n) (-1 + a + 2 e + 2 n) (a + 2 e + 2 n) (-1 + c + 2 e + 2 n) (c + 2 e + 2 n) (-1 + 2 a - 2 b - 2 d + 2 e + 2 n) (-1 + 2 b + 4 e + 4 n) Pochhammer[1/2 + a - b - c - d + e, n] Pochhammer[1 + 2 e, 2 n] Pochhammer[1/2 + a - b + 2 e, 2 n] Pochhammer[1 + a - c + 2 e, 2 n] Pochhammer[ 1 + a - d + 3 e, 3 n])), {n, 1, Infinity}] Sum[((2 + a + 4 e + 2 k) (1 + 2 b + 4 e + 2 k) Pochhammer[1 + d + e, k] Pochhammer[1 + a + 2 e, k] Pochhammer[1/2 + b + 2 e, k] Pochhammer[1 + c + 2 e, k])/((1 + 2 e + k) (1 + a - c + 2 e + k) (1 + a - d + 3 e + k) (1 + 2 a - 2 b + 4 e + 2 k) Pochhammer[1 + 2 e, k] Pochhammer[1/2 + a - b + 2 e, k] Pochhammer[ 1 + a - c + 2 e, k] Pochhammer[1 + a - d + 3 e, k]), {k, 0, Infinity}]]

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Second formula, in copyable Mathematica format:

TraditionalForm[-4*(b - d) (-a - b + c + d) HoldForm@ Sum[((-1)^ n (-a^2 - a b - 2 a^2 b - 2 a b^2 + 3 a c + 2 a^2 c + 2 b c + 8 a b c + 4 b^2 c - 2 c^2 - 6 a c^2 - 8 b c^2 + 4 c^3 + 3 a d + 4 a^2 d + 3 b d + 12 a b d + 6 b^2 d - 5 c d - 18 a c d - 24 b c d + 18 c^2 d - 4 d^2 - 14 a d^2 - 18 b d^2 + 28 c d^2 + 14 d^3 + 3 a n + 4 a^2 n + 3 b n + 12 a b n + 6 b^2 n - 5 c n - 18 a c n - 24 b c n + 18 c^2 n - 8 d n - 28 a d n - 36 b d n + 56 c d n + 42 d^2 n - 4 n^2 - 14 a n^2 - 18 b n^2 + 28 c n^2 + 42 d n^2 + 14 n^3) Pochhammer[1 - a + d, n] Pochhammer[1 + a + d, n] Pochhammer[1 - b + d, n] Pochhammer[1/2 - b + c + d, n] Pochhammer[ 1 - a - b + c + d, n] Pochhammer[1 - a + 2 c + d, n])/((a + d + n) (-b + d + n) (-a - b + c + d + n) (-a + 2 c + d + n) (-1 - 2 b + 2 c + 2 d + 2 n) Pochhammer[1 + d, n] Pochhammer[1/2 + c + d, n] Pochhammer[1 - a + c + d, n] Pochhammer[1 - a - 2 b + 2 c + 3 d, 3 n]), {n, 1, Infinity}] == (1 - a + d) (b - d)* HoldForm@ Sum[(2^(1 - 2 k) Pochhammer[1 + b, k] Pochhammer[1 + a + d, 2 k])/((1 + d + k) (1 + 2 c + 2 d + 2 k) Pochhammer[ 1 + a + b - c - d, k] Pochhammer[1 + d, k] Pochhammer[ 1/2 + c + d, k]), {k, 0, Infinity}] + ( 2^(1 + a + d) Gamma[1 + a + b - c - d] Gamma[1 + d] Gamma[1/2 + c + d])/( Sqrt[\[Pi]] Gamma[1 + b] Gamma[1 + a + d]) (1 + HoldForm@ Sum[-((2^(-1 - 2 k) (a + 2 b - 2 c - 3 d) (1 - a + 2 c + d) Pochhammer[ 1 + a + 2 b - 2 c - 3 d, 2 k])/((1 + b - d + k) (1 + 2 b - 2 c - 2 d + 2 k) Pochhammer[1 + b - d, k] Pochhammer[1/2 + b - c - d, k])), {k, 0, Infinity}]) - (( 2^(1 + 2 b) Gamma[1 + b - d] Gamma[1/2 + b - c - d] Gamma[ 1 + a + b - c - d] Gamma[1 + d] Gamma[1/2 + c + d] Gamma[ 1 - a + c + d] Gamma[1 - a - 2 b + 2 c + 3 d])/(\[Pi] Gamma[ 1 + b] Gamma[1 - a + d] Gamma[1 + a + d] Gamma[ 1 - a + 2 c + d]))]