A141153 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) ).

1, 1, 3, 31, 1609, 626097, 2407996027, 110977327013551, 71594581089754557777, 738994182797188307880872353, 137301106425308220881681919632979379, 510195974626378486585193070538567102152265599

Offset

2,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 2..41

Formula

a(n+1) = n!*Sum_{k=0..n} 2^(n-k)/k!*a(k)*a(n-k), (offset 0). - Vladeta Jovovic, Jul 08 2008

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]

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

Example

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 +...
exp(A(x)) = 1 + x^2/2 + 3*x^4/8 + 31*x^6/48 + 1609*x^8/384 + 626097*x^10/3840 +...
Contribution from Paul D. Hanna, Aug 09 2009: (Start)
E.g.f.: E(x) = 1 + x + 3x^2/2! + 31*x^3/3! + 1609*x^4/4! +...(offset 0);
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)

Mathematica

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 *)

Prog

(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))}
Contribution from Paul D. Hanna, Aug 09 2009: (Start)
(PARI) /* E.g.f. exp(Sum_{n>=1} 2^(n-1)*a(n-1)*x^n/n) with offset 0: */
{a(n)=n!*polcoeff(exp(sum(m=1, n, 2^(m-1)*a(m-1)*x^m/m)+x*O(x^n)), n)} (End)

Crossrefs

Cf. A000178, A002109.

Sequence in context: A328811 A136584 A194937 * A144906 A081789 A066976

Adjacent sequences: A141150 A141151 A141152 * A141154 A141155 A141156

Keyword

nonn

Author

Paul D. Hanna, Jun 11 2008

Status

approved