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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
1, 7, 64, 667, 7526, 89614, 1109578, 14153379, 184819348, 2459689862, 33253032748, 455530364830, 6310982029730, 88288166501864, 1245647703839594, 17706547056186467, 253368343687134676, 3647065046069378674, 52777288671194300790, 767433117054617825162 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 13 2012
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)