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Alternating sums of the cubes of the central binomial coefficients.
1

%I #24 Jul 07 2020 08:01:12

%S 1,7,209,7791,335209,15667799,773221225,39651016343,2092095886657,

%T 112840936041343,6193764391911873,344853399798469695,

%U 19429178297906958721,1105629520934309041279,63455683531507986958721,3668895994183490904049279

%N Alternating sums of the cubes of the central binomial coefficients.

%H Robert Israel, <a href="/A275822/b275822.txt">Table of n, a(n) for n = 0..555</a>

%H The Wolfram Functions Site, <a href="http://functions.wolfram.com/EllipticIntegrals/EllipticK/introductions/CompleteEllipticIntegrals/02/">Complete Elliptic Integrals</a>

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2*k,k)^3.

%F 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 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 a(n) ~ 2^(6*n+6) / (65*Pi^(3/2)*n^(3/2)). - _Vaclav Kotesovec_, Nov 16 2016

%p L:= [seq((-1)^k*binomial(2*k,k)^3,k=0..20)]:

%p B:= ListTools:-PartialSums(L):

%p seq((-1)^(k+1)*B[k],k=1..nops(B)); # _Robert Israel_, Nov 21 2016

%t Table[Sum[Binomial[2 k, k]^3 (-1)^(n - k), {k, 0, n}], {n, 0, 20}]

%t Table[Sum[(-1)^(n - k) (k + 1)^3 CatalanNumber[k]^3, {k, 0, n}], {n, 0, 20}] (* _Jan Mangaldan_, Jul 07 2020 *)

%o (Maxima) makelist(sum(binomial(2*k,k)^3*(-1)^(n-k),k,0,n),n,0,12);

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k,k)^3); \\ _Michel Marcus_, Jul 07 2020

%Y Cf. A079727.

%K nonn

%O 0,2

%A _Emanuele Munarini_, Nov 15 2016