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A213402 Expansion of exp( Sum_{n>=1} binomial(2*n^2-1, n^2) * x^n/n ). 2
1, 1, 18, 8121, 75143314, 12641135924840, 36876057665515782388, 1819829483946334379705083199, 1496946629440335290077827171482478552, 20328210229221517747154723779733726877327901902, 4527425732825492268524878479039840102127714838348518156649 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Self-convolution square-root of A201556.

Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108): C(x) = exp( Sum_{n>=1} binomial(2*n-1,n) * x^n/n ).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..40

FORMULA

a(n) = (1/n) * Sum_{k=1..n} C(2*k^2,k^2)/2 * a(n-k) for n>0 with a(0)=1.

a(n) ~ 2^(2*n^2-1) / (sqrt(Pi)*n^2). - Vaclav Kotesovec, Mar 06 2014

EXAMPLE

G.f.: A(x) = 1 + x + 18*x^2 + 8121*x^3 + 75143314*x^4 + 12641135924840*x^5 +...

where

log(A(x)) = x + 35*x^2/2 + 24310*x^3/3 + 300540195*x^4/4 +...+ C(2*n^2-1,n^2)*x^n/n +...

MATHEMATICA

nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[2*k^2, k^2]/2*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m^2, m^2)/2*x^m/m)+x*O(x^n)), n)}

(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, binomial(2*k^2, k^2)/2*a(n-k)))}

for(n=0, 21, print1(a(n), ", "))

CROSSREFS

Cf. A201556, A200002.

Sequence in context: A232595 A159405 A265450 * A248804 A060617 A222202

Adjacent sequences:  A213399 A213400 A213401 * A213403 A213404 A213405

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 10 2012

STATUS

approved

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Last modified April 19 03:00 EDT 2019. Contains 322237 sequences. (Running on oeis4.)