cp's OEIS Frontend

A276537 Alternating binomial sums of the cubes of the central binomial coefficients.

%I A276537 #39 Mar 07 2025 06:43:16
%S A276537 1,7,201,7375,312265,14365887,697859169,35226348087,1829569294665,
%T A276537 97138289500735,5248514415816721,287657066913117447,
%U A276537 15953440327189548001,893653778439275931175,50488236061157830951545,2873526763346873838886815
%N A276537 Alternating binomial sums of the cubes of the central binomial coefficients.
%H A276537 Seiichi Manyama, <a href="/A276537/b276537.txt">Table of n, a(n) for n = 0..558</a>
%H A276537 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html">Complete Elliptic Integral of the First Kind</a>
%H A276537 The Wolfram Functions Site, <a href="http://functions.wolfram.com/EllipticIntegrals/EllipticK/introductions/CompleteEllipticIntegrals/02/">Complete Elliptic Integrals</a>, 2016.
%F A276537 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*binomial(2*k,k)^3.
%F A276537 Recurrence: (n^3+12*n^2+48*n+64)*a(n+4)-(60*n^3+630*n^2+2204*n+2569)*a(n+3)-(186*n^3+1674*n^2+5037*n+5067)*a(n+2)-94*(2*n^3+15*n^2+37*n+30)*a(n+1)-63*(n^3+6*n^2+11*n+6)*a(n)=0.
%F A276537 G.f.: (4/Pi^2)*K(1/2-1/2*sqrt((1-63*t)/(1+t)))^2/(1+t), where K(x) is the complete elliptic integral of the first kind (defined as in MathWorld or in The Wolfram Functions Site).
%F A276537 a(n) ~ 3^(2*n+3) * 7^(n+3/2) / (512 * Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Nov 16 2016
%F A276537 a(n) = (-1)^n*4F3(1/2,1/2,1/2,-n; 1,1,1; 64). - _Ilya Gutkovskiy_, Nov 25 2016
%t A276537 Table[Sum[Binomial[n,k]Binomial[2k,k]^3(-1)^(n-k),{k,0,n}],{n,0,100}]
%o A276537 (Maxima) makelist(sum(binomial(n,k)*binomial(2*k,k)^3*(-1)^(n-k),k,0,n),n,0,12);
%o A276537 (Magma) [&+[(-1)^(n-k)*Binomial(n,k)*Binomial(2*k,k)^3: k in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Dec 03 2016
%Y A276537 Cf. A276536.
%Y A276537 Similar sums of m-powers of the central binomial coefficients: A002426 (m=1), A278934 (m=2), this sequence (m=3).
%K A276537 nonn
%O A276537 0,2
%A A276537 _Emanuele Munarini_, Nov 16 2016