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%I #12 Mar 01 2014 17:08:26
%S 1,1,3,31,1609,626097,2407996027,110977327013551,71594581089754557777,
%T 738994182797188307880872353,137301106425308220881681919632979379,
%U 510195974626378486585193070538567102152265599
%N G.f.: A(x) = Sum_{n>=1} a(n-1)*x^(2*n)/(2*n) = log( Sum_{n>=0} a(n)*x^(2*n)/(n!*2^n) ).
%H Vaclav Kotesovec, <a href="/A141153/b141153.txt">Table of n, a(n) for n = 2..41</a>
%F a(n+1) = n!*Sum_{k=0..n} 2^(n-k)/k!*a(k)*a(n-k), (offset 0). - _Vladeta Jovovic_, Jul 08 2008
%F E.g.f.: Sum_{n>=0} a(n)*x^n/n! = exp( Sum_{n>=1} 2^(n-1)*a(n-1)*x^n/n ) (offset 0). [From _Paul D. Hanna_, Aug 09 2009]
%F a(n) ~ c * 2^(n*(n-4)/2) * Pi^(n/2) * n^((n-2)^2/2 - 1/12) / exp(n*(3*n-8)/4), where c = 2.294946359935163474113719941809113139554600453... - _Vaclav Kotesovec_, Feb 27 2014
%e G.f.: A(x) = x^2/2 + x^4/4 + 3*x^6/6 + 31*x^8/8 + 1609*x^10/10 + 626097*x^12/12 +...
%e exp(A(x)) = 1 + x^2/2 + 3*x^4/8 + 31*x^6/48 + 1609*x^8/384 + 626097*x^10/3840 +...
%e Contribution from _Paul D. Hanna_, Aug 09 2009: (Start)
%e E.g.f.: E(x) = 1 + x + 3x^2/2! + 31*x^3/3! + 1609*x^4/4! +...(offset 0);
%e E(x) = exp(1*x + 1*2*x^2/2 + 3*2^2*x^3/3 + 31*2^3*x^4/4 + 1609*2^4*x^5/5 +...) (End)
%t nmax = 20; b = ConstantArray[0, nmax+2]; b[[1]] = 1; b[[2]] = 1; Do[b[[n+2]] = n!*Sum[2^(n-k)/k!*b[[k+1]]*b[[n-k+1]], {k, 0, n}], {n, 1, nmax}]; b (* _Vaclav Kotesovec_, Feb 27 2014 *)
%o (PARI) {a(n)=if(n==0, 1, n!*2^n*polcoeff(exp(sum(k=0, n-1, a(k)*x^(2*k+2)/(2*k+2))+O(x^(2*n+2))),2*n))}
%o Contribution from _Paul D. Hanna_, Aug 09 2009: (Start)
%o (PARI) /* E.g.f. exp(Sum_{n>=1} 2^(n-1)*a(n-1)*x^n/n) with offset 0: */
%o {a(n)=n!*polcoeff(exp(sum(m=1,n,2^(m-1)*a(m-1)*x^m/m)+x*O(x^n)),n)} (End)
%Y Cf. A000178, A002109.
%K nonn
%O 2,3
%A _Paul D. Hanna_, Jun 11 2008
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