A307810 - OEIS

%I #23 May 13 2021 02:35:36

%S 1,100,13924,2371600,453093796,92598490000,19745403216400,

%T 4333667896360000,971177275449892900,221106619001508490000,

%U 50967394891692703241104,11866732390447357481358400,2785834789480617203561744656,658549235163074008904405646400

%N Expansion of 1/AGM(1-64*x, sqrt((1-16*x)*(1-256*x))).

%C See A246923.

%C Also the squares of coefficients in g.f. 1/sqrt((1-4*x)*(1-16*x)).

%H Seiichi Manyama, <a href="/A307810/b307810.txt">Table of n, a(n) for n = 0..416</a>

%F a(n) = A307695(n)^2 = (Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k))^2.

%F a(n) ~ 2^(8*n+2) / (3*Pi*n). - _Vaclav Kotesovec_, Sep 27 2019

%t a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 14, 0] // Flatten (* _Amiram Eldar_, May 13 2021 *)

%o (PARI) N=20; x='x+O('x^N); Vec(1/agm(1-64*x, sqrt((1-16*x)*(1-256*x))))

%o (PARI) {a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))^2}

%o (PARI) {a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))^2}

%Y Cf. A307695.

%Y (Sum_{k=0..n} c^(n-k)*e^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} d^(n-k)*(-e)^k*binomial(n,k)*binomial(2k,k))^2, where e = (d-c)/4: A002894 (c=0,d=4,e=1), A246467 (c=1,d=5,e=1), A246876 (c=2,d=6,e=1), A246906 (c=3,d=7,e=1), A307811 (c=5,d=9,e=1), A322240 (c=-3,d=5,e=2), A322243 (c=-1,d=7,e=2), A246923 (c=1,d=9,e=2), A248167 (c=3, d=11,e=2), A322247 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 30 2019