cp's OEIS Frontend

A275822 Alternating sums of the cubes of the central binomial coefficients.

%I A275822 #24 Jul 07 2020 08:01:12
%S A275822 1,7,209,7791,335209,15667799,773221225,39651016343,2092095886657,
%T A275822 112840936041343,6193764391911873,344853399798469695,
%U A275822 19429178297906958721,1105629520934309041279,63455683531507986958721,3668895994183490904049279
%N A275822 Alternating sums of the cubes of the central binomial coefficients.
%H A275822 Robert Israel, <a href="/A275822/b275822.txt">Table of n, a(n) for n = 0..555</a>
%H A275822 The Wolfram Functions Site, <a href="http://functions.wolfram.com/EllipticIntegrals/EllipticK/introductions/CompleteEllipticIntegrals/02/">Complete Elliptic Integrals</a>
%F A275822 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2*k,k)^3.
%F A275822 Recurrence: (n+2)^3*a(n+2)-(3*n+4)*(21*n^2+66*n+52)*a(n+1)-8*(2n+3)^3*a(n)=0.
%F A275822 G.f.: (4/Pi^2)*K((1-sqrt(1-64*t))/2)^2/(1+t), where K(x) is complete elliptic integral of the first kind (defined as in The Wolfram Functions Site).
%F A275822 a(n) ~ 2^(6*n+6) / (65*Pi^(3/2)*n^(3/2)). - _Vaclav Kotesovec_, Nov 16 2016
%p A275822 L:= [seq((-1)^k*binomial(2*k,k)^3,k=0..20)]:
%p A275822 B:= ListTools:-PartialSums(L):
%p A275822 seq((-1)^(k+1)*B[k],k=1..nops(B)); # _Robert Israel_, Nov 21 2016
%t A275822 Table[Sum[Binomial[2 k, k]^3 (-1)^(n - k), {k, 0, n}], {n, 0, 20}]
%t A275822 Table[Sum[(-1)^(n - k) (k + 1)^3 CatalanNumber[k]^3, {k, 0, n}], {n, 0, 20}] (* _Jan Mangaldan_, Jul 07 2020 *)
%o A275822 (Maxima) makelist(sum(binomial(2*k,k)^3*(-1)^(n-k),k,0,n),n,0,12);
%o A275822 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k,k)^3); \\ _Michel Marcus_, Jul 07 2020
%Y A275822 Cf. A079727.
%K A275822 nonn
%O A275822 0,2
%A A275822 _Emanuele Munarini_, Nov 15 2016