A213515 L.g.f.: log( Sum_{n>=0} A000108(n)^2*x^n ) = Sum_{n>=1} a(n)*x^n/n, where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.

1, 7, 64, 667, 7526, 89614, 1109578, 14153379, 184819348, 2459689862, 33253032748, 455530364830, 6310982029730, 88288166501864, 1245647703839594, 17706547056186467, 253368343687134676, 3647065046069378674, 52777288671194300790, 767433117054617825162

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..830

Formula

a(n) ~ 2^(4*n - 2) / ((4 - Pi) * n^2). - Vaclav Kotesovec, Oct 05 2019

Example

L.g.f.: L(x) = x + 7*x^2/2 + 64*x^3/3 + 667*x^4/4 + 7526*x^5/5 + 89614*x^6/6 +...
such that
exp(L(x)) = 1 + x + 2^2*x^2 + 5^2*x^3 + 14^2*x^4 + 42^2*x^5 + 132^2*x^6 + 429^2*x^7 +...+ A000108(n)^2*x^n +...
G.f: (Pi - 3*E(4*sqrt(x)) + (1-16*x)*K(4*sqrt(x)))/(4*E(4*sqrt(x)) - 2*(1-16*x)*K(4*sqrt(x)) - Pi), where K(x) and E(x) are the complete elliptic integrals of the 1st and 2nd kind. - Vladimir Reshetnikov, Nov 11 2015

Mathematica

Series[(Pi - 3 EllipticE[16 x] + (1 - 16 x) EllipticK[16 x])/(4 EllipticE[16 x] - 2 (1 - 16 x) EllipticK[16 x] - Pi), {x, 0, 20}][[3]] (* Vladimir Reshetnikov, Nov 11 2015 *)

Prog

(PARI) {a(n)=n*polcoeff(log(sum(m=0, n+1, binomial(2*m, m)^2/(m+1)^2*x^m)+x*O(x^n)), n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A001246.

Sequence in context: A288690 A362726 A371404 * A349333 A293470 A256506

Adjacent sequences: A213512 A213513 A213514 * A213516 A213517 A213518

Keyword

nonn

Author

Paul D. Hanna, Jun 13 2012

Status

approved