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A307695 Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).

%I A307695 #36 May 13 2021 02:36:24
%S A307695 1,10,118,1540,21286,304300,4443580,65830600,985483270,14869654300,
%T A307695 225759595348,3444812388280,52781007848284,811510465220920,
%U A307695 12513859077134008,193460383702061200,2997463389599395270,46532910920993515900,723626591914643806180,11270311875128088314200
%N A307695 Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).
%C A307695 Let 1/(sqrt(1-c*x)*sqrt(1-d*x)) = Sum_{k>=0} b(k)*x^k.
%C A307695 b(n) = Sum_{k=0..n} c^(n-k) * e^k * binomial(n,k) * binomial(2*k,k) = Sum_{k=0..n} d^(n-k) * (-e)^k * binomial(n,k) * binomial(2*k,k), where e = (d-c)/4.
%C A307695 n*b(n) = (c+d)/2 * (2*n-1) * b(n-1) - c * d * (n-1) * b(n-2) for n > 1.
%H A307695 Seiichi Manyama, <a href="/A307695/b307695.txt">Table of n, a(n) for n = 0..831</a>
%F A307695 a(n) = Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k).
%F A307695 a(n) = Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k).
%F A307695 D-finite with recurrence: n*a(n) = 10*(2*n-1)*a(n-1) - 64*(n-1)*a(n-2) for n > 1.
%F A307695 a(n) ~ 2^(4*n+1) / sqrt(3*Pi*n). - _Vaclav Kotesovec_, Apr 30 2019
%t A307695 a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* _Amiram Eldar_, May 13 2021 *)
%o A307695 (PARI) N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))
%o A307695 (PARI) {a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}
%o A307695 (PARI) {a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}
%Y A307695 Cf. A000984 (c=0,d=4,e=1), A026375 (c=1,d=5,e=1), A081671 (c=2,d=6,e=1), A098409 (c=3,d=7,e=1), A098410 (c=4,d=8,e=1), A104454 (c=5,d=9,e=1).
%Y A307695 Cf. A084605 (c=-3,d=5,e=2), A098453 (c=-2,d=6,e=2), A322242 (c=-1,d=7,e=2), A084771 (c=1,d=9,e=2), A248168 (c=3,d=11,e=2).
%Y A307695 Cf. A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
%Y A307695 Cf. A322244 (c=-5,d=11,e=4), A322248 (c=-3,d=13,e=4).
%K A307695 nonn
%O A307695 0,2
%A A307695 _Seiichi Manyama_, Apr 22 2019