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A307811 Expansion of 1/AGM(1-45*x, sqrt((1-25*x)*(1-81*x))).

%I A307811 #20 May 13 2021 02:35:31
%S A307811 1,49,2601,148225,8970025,570111129,37678303881,2567836387809,
%T A307811 179267329355625,12754414737348025,921185098227422161,
%U A307811 67340346156989933769,4971327735657992896201,369994703739586257235225,27725052308247030792515625,2089567204521186409129541025
%N A307811 Expansion of 1/AGM(1-45*x, sqrt((1-25*x)*(1-81*x))).
%C A307811 See A246923.
%C A307811 Also the squares of coefficients in g.f. 1/sqrt((1-5*x)*(1-9*x)).
%H A307811 Seiichi Manyama, <a href="/A307811/b307811.txt">Table of n, a(n) for n = 0..525</a>
%F A307811 a(n) = A104454(n)^2 = (Sum_{k=0..n} 5^(n-k)*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 9^(n-k)*(-1)^k*binomial(n,k)*binomial(2k,k))^2.
%F A307811 a(n) ~ 3^(4*n+2) / (4*Pi*n). - _Vaclav Kotesovec_, Sep 27 2019
%t A307811 a[n_] := Sum[5^(n-k) * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 16, 0] // Flatten (* _Amiram Eldar_, May 13 2021 *)
%o A307811 (PARI) N=20; x='x+O('x^N); Vec(1/agm(1-45*x, sqrt((1-25*x)*(1-81*x))))
%o A307811 (PARI) {a(n) = sum(k=0, n, 5^(n-k)*binomial(n, k)*binomial(2*k, k))^2}
%o A307811 (PARI) {a(n) = sum(k=0, n, 9^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))^2}
%Y A307811 Cf. A104454.
%Y A307811 (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), this sequence (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), A307810 (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).
%K A307811 nonn
%O A307811 0,2
%A A307811 _Seiichi Manyama_, Apr 30 2019