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A276536 Binomial sums of the cubes of the central binomial coefficients.

%I A276536 #34 Mar 07 2025 06:43:20
%S A276536 1,9,233,8673,376329,17800209,890215361,46294813497,2478150328777,
%T A276536 135642353562321,7556884938829233,427106589765940137,
%U A276536 24429206859151618209,1411391470651692285609,82245902444586364980057,4828398428680134702936273
%N A276536 Binomial sums of the cubes of the central binomial coefficients.
%H A276536 Seiichi Manyama, <a href="/A276536/b276536.txt">Table of n, a(n) for n = 0..554</a>
%H A276536 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html">Complete Elliptic Integral of the First Kind</a>
%H A276536 The Wolfram Functions Site, <a href="http://functions.wolfram.com/EllipticIntegrals/EllipticK/introductions/CompleteEllipticIntegrals/02/">Complete Elliptic Integrals</a>, 2016.
%F A276536 a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(2*k, k)^3.
%F A276536 Recurrence: (n^3 + 12n^2 + 48n + 64) * a(n+4) - (68n^3 + 714n^2 + 2500n + 2919) * a(n+3) + (198n^3 + 1782n^2 + 5363n + 5397) * a(n+2) - 98 * (2n^3 + 15n^2 + 37n + 30) * a(n+1) + 65 * (n^3 + 6n^2 + 11n + 6) * a(n) = 0.
%F A276536 G.f.: (4/Pi^2) * K(1/2 - 1/2 * sqrt((1-65*t)/(1-t)))^2 / (1-t), where K(x) is complete elliptic integral of the first kind (defined as in MathWorld or in The Wolfram Functions Site).
%F A276536 a(n) ~ 65^(n+3/2) / (512 * Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Nov 16 2016
%F A276536 a(n) = 4F3(1/2,1/2,1/2,-n; 1,1,1; -64). - _Ilya Gutkovskiy_, Nov 25 2016
%t A276536 Table[Sum[Binomial[n, k]Binomial[2k, k]^3, {k, 0, n}], {n, 0, 100}]
%o A276536 (Maxima) makelist(sum(binomial(n,k)*binomial(2*k,k)^3,k,0,n),n,0,12);
%o A276536 (Magma) [&+[Binomial(n, k)*Binomial(2*k, k)^3: k in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Nov 30 2016
%Y A276536 Cf. Sum_{k = 0..n} binomial(n, k)*binomial(2*k, k)^m: A026375 (m=1), A248586 (m=2), this sequence (m=3).
%K A276536 nonn
%O A276536 0,2
%A A276536 _Emanuele Munarini_, Nov 16 2016