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A213402 Expansion of exp( Sum_{n>=1} binomial(2*n^2-1, n^2) * x^n/n ).

%I A213402 #19 Mar 31 2017 03:53:14
%S A213402 1,1,18,8121,75143314,12641135924840,36876057665515782388,
%T A213402 1819829483946334379705083199,1496946629440335290077827171482478552,
%U A213402 20328210229221517747154723779733726877327901902,4527425732825492268524878479039840102127714838348518156649
%N A213402 Expansion of exp( Sum_{n>=1} binomial(2*n^2-1, n^2) * x^n/n ).
%C A213402 Self-convolution square-root of A201556.
%C A213402 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 ).
%H A213402 G. C. Greubel, <a href="/A213402/b213402.txt">Table of n, a(n) for n = 0..40</a>
%F A213402 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.
%F A213402 a(n) ~ 2^(2*n^2-1) / (sqrt(Pi)*n^2). - _Vaclav Kotesovec_, Mar 06 2014
%e A213402 G.f.: A(x) = 1 + x + 18*x^2 + 8121*x^3 + 75143314*x^4 + 12641135924840*x^5 +...
%e A213402 where
%e A213402 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 +...
%t A213402 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 *)
%o A213402 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m^2, m^2)/2*x^m/m)+x*O(x^n)), n)}
%o A213402 (PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, binomial(2*k^2, k^2)/2*a(n-k)))}
%o A213402 for(n=0,21,print1(a(n),", "))
%Y A213402 Cf. A201556, A200002.
%K A213402 nonn
%O A213402 0,3
%A A213402 _Paul D. Hanna_, Jun 10 2012