A156326 E.g.f.: A(x) = exp( Sum_{n>=1} n^2 * a(n-1)*x^n/n! ) = Sum_{n>=0} a(n)*x^n/n! with a(0) = 1.

1, 1, 5, 58, 1181, 36696, 1601497, 92969920, 6908883417, 638746871680, 71860612355981, 9664570175364864, 1531263494465900725, 282321785979644121088, 59935663751282958139425, 14517627118656645274771456, 3980008380007702720451029553, 1226189930561023692489563013120

Offset: 0

Author

Paul D. Hanna, Feb 08 2009

Keywords

• nonn

Examples

			E.g.f: A(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! + ...
log(A(x)) = x + 2^2*x^2/2! + 3^2*5*x^3/3! + 4^2*58*x^4/4! + 5^2*1181*x^5/5! + ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A156325, A156327, A182962, A161968.

Programs

Mathematica

nmax = 20; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = Sum[k^2 * Binomial[n-1,k-1]*b[[k]]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b (* Vaclav Kotesovec, Feb 27 2014 *)

PARI

{a(n)=if(n==0,1,n!*polcoeff(exp(sum(k=1,n,k^2*a(k-1)*x^k/k!)+x*O(x^n)),n))}

PARI

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

PARI

seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n] = sum(k=1, n, k^2 * binomial(n-1,k-1)*a[k]*a[1+n-k])); a} \\ Andrew Howroyd, Jan 05 2020

Formula

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

E.g.f.: A(x) = exp( x*A(x) + x^2*A'(x) ). - Paul D. Hanna, Apr 02 2018

E.g.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x/G(x)) = G(x) is the e.g.f. of A182962, which satisfies:

. G(x) = exp( x/(1 - x*G'(x)/G(x)) );

. a(n) = [x^n/n!] G(x)^(n+1)/(n+1) for n>=0.

a(n) = A161968(n+1)/(n+1), where L(x) = x*exp(x*d/dx L(x)) is the e.g.f. of A161968. - Paul D. Hanna, Feb 21 2014

a(n) ~ c * n * (n!)^2, where c = A238223 * exp(1) = 0.592451670452494179138706062417512405957... - Vaclav Kotesovec, Feb 27 2014

Extensions

Terms a(15) and beyond from Andrew Howroyd, Jan 05 2020