A156327 E.g.f.: A(x) = exp( Sum_{n>=1} n*(n+3)/2 * a(n-1)*x^n/n! ) = Sum_{n>=0} a(n)*x^n/n! with a(0)=1.

1, 2, 14, 194, 4280, 134232, 5587408, 294882464, 19102334112, 1482726089600, 135370060595264, 14325189014356992, 1736329123715436544, 238698935851482530816, 36911830664814417907200, 6375425555384677316100608, 1222423907917065757088181248, 258802786174190320917263867904

Offset: 0

Author

Paul D. Hanna, Feb 08 2009

Keywords

• nonn

Examples

			E.g.f: A(x) = 1 + 2*x + 14*x^2/2! + 194*x^3/3! + 4280*x^4/4! + 134232*x^5/5! +...
log(A(x)) = 2*1*x + 5*2*x^2/2! + 9*14*x^3/3! + 14*194*x^4/4! + 20*4280*x^5/5! +...
		

Crossrefs

Cf. A156325, A156326.

Programs

Mathematica

terms = 18; A[] = 1; Do[A[x] = Exp[2x*A[x]+x^2*A'[x]/2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)

PARI

{a(n)=if(n==0,1,n!*polcoeff(exp(sum(k=1,n,k*(k+3)/2*a(k-1)*x^k/k!)+x*O(x^n)),n))}

PARI

{a(n)=if(n==0,1,sum(k=1,n,k*(k+3)/2*binomial(n-1,k-1)*a(k-1)*a(n-k)))}

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(2+j/2)*binomial(i-1, j)*v[j+1]*v[i-j])); v; \\ Seiichi Manyama, Jul 25 2025

Formula

a(n) = Sum_{k=1..n} k*(k+3)/2 * C(n-1,k-1)*a(k-1)*a(n-k) for n>0, with a(0)=1.

E.g.f. A(x) satisfies A(x) = exp(2 * x * A(x) + x^2/2 * A'(x)). - Seiichi Manyama, Jul 25 2025

a(n) ~ c * n!^2 * n^7 / 2^n, where c = 0.00029014625163457216349268... - Vaclav Kotesovec, Aug 05 2025