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%I #14 Oct 05 2019 09:15:49
%S 1,7,64,667,7526,89614,1109578,14153379,184819348,2459689862,
%T 33253032748,455530364830,6310982029730,88288166501864,
%U 1245647703839594,17706547056186467,253368343687134676,3647065046069378674,52777288671194300790,767433117054617825162
%N 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.
%H Vaclav Kotesovec, <a href="/A213515/b213515.txt">Table of n, a(n) for n = 1..830</a>
%F a(n) ~ 2^(4*n - 2) / ((4 - Pi) * n^2). - _Vaclav Kotesovec_, Oct 05 2019
%e 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 +...
%e such that
%e 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 +...
%e 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
%t 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 *)
%o (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)}
%o for(n=1,25,print1(a(n),", "))
%Y Cf. A001246.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 13 2012
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