Example 1: F(n,k) is

((-1)^n (4 k + 6 n + 1) Gamma[1/2 + n]^10 Gamma[1/2 + k + n]^5 Gamma[k + 4 n])/(Gamma[2 n]^5* Gamma[1 + 2 n + k]^5* Gamma[k - n + 3/2])

G(n,k)/F(n,k) is 

1/(131072 n^5 (1 + k + 2 n)^5 (1 + 4 k + 6 n))(-1 - 2 k + 2 n) (42 k + 443 k^2 + 1955 k^3 + 4666 k^4 + 6488 k^5 + 5248 k^6 + 2288 k^7 + 416 k^8 + 168 n + 3124 k n + 20830 k^2 n + 70646 k^3 n + 137320 k^4 n + 158672 k^5 n + 107104 k^6 n + 38752 k^7 n + 5760 k^8 n + 5408 n^2 + 71720 k n^2 + 377316 k^2 n^2 + 1044300 k^3 n^2 + 1671160 k^4 n^2 + 1581888 k^5 n^2 + 859968 k^6 n^2 + 242240 k^7 n^2 + 26240 k^8 n^2 + 78720 n^3 + 846256 k n^3 + 3648280 k^2 n^3 + 8226200 k^3 n^3 + 10507200 k^4 n^3 + 7611264 k^5 n^3 + 2896000 k^6 n^3 + 446080 k^7 n^3 + 674816 n^4 + 5969760 k n^4 + 20818480 k^2 n^4 + 36867600 k^3 n^4 + 35094880 k^4 n^4 + 17069440 k^5 n^4 + 3321600 k^6 n^4 + 3700480 n^5 + 26329664 k n^5 + 71426976 k^2 n^5 + 92990240 k^3 n^5 + 58335872 k^4 n^5 + 14148864 k^5 n^5 + 13177856 n^6 + 72457088 k n^6 + 143428800 k^2 n^6 + 121932352 k^3 n^6 + 37706880 k^4 n^6 + 30078976 n^7 + 120101120 k n^7 + 154486912 k^2 n^7 + 64371840 k^3 n^7 + 42270720 n^8 + 109340672 k n^8 + 68728320 k^2 n^8 + 33236992 n^9 + 41932800 k n^9 + 11182080 n^10)
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Example 2: F(n,k) is

((-1)^n (1/2 + 2 k + 3 n) Gamma[1/2 + n]^6 Gamma[1/2 + k + n]^4 Gamma[ 1/2 + k + 2 n] Gamma[k + 3 n])/( Gamma[3/2 + k] Gamma[n]^4 Gamma[2 n] Gamma[1 + k + n] Gamma[ 1 + k + 2 n]^4)

G(n,k)/F(n,k) is

-1/(1024 n^5 (1 + k + 2 n)^4 (1/2 + 2 k + 3 n))(1 + 2 k) (3 k + 29 k^2 + 114 k^3 + 232 k^4 + 256 k^5 + 144 k^6 + 32 k^7 + 9 n + 160 k n + 964 k^2 n + 2816 k^3 n + 4432 k^4 n + 3776 k^5 n + 1600 k^6 n + 256 k^7 n + 219 n^2 + 2654 k n^2 + 12164 k^2 n^2 + 27720 k^3 n^2 + 33776 k^4 n^2 + 21600 k^5 n^2 + 6464 k^6 n^2 + 640 k^7 n^2 + 2364 n^3 + 22336 k n^3 + 80560 k^2 n^3 + 142080 k^3 n^3 + 128448 k^4 n^3 + 55808 k^5 n^3 + 8960 k^6 n^3 + 14772 n^4 + 110680 k n^4 + 308720 k^2 n^4 + 400480 k^3 n^4 + 240704 k^4 n^4 + 53376 k^5 n^4 + 58144 n^5 + 336512 k n^5 + 686272 k^2 n^5 + 584704 k^3 n^5 + 175360 k^4 n^5 + 146000 n^6 + 614816 k n^6 + 817344 k^2 n^6 + 343424 k^3 n^6 + 225984 n^7 + 616960 k n^7 + 401664 k^2 n^7 + 195776 n^8 + 260480 k n^8 + 72448 n^9)