A161968 E.g.f. L(x) satisfies: L(x) = x*exp(x*d/dx L(x)), where L(x) is the logarithm of e.g.f. of A161967.

1, 2, 15, 232, 5905, 220176, 11210479, 743759360, 62179950753, 6387468716800, 790466735915791, 115974842104378368, 19906425428056709425, 3952505003715017695232, 899034956269244372091375, 232282033898506324396343296, 67660142460130946247667502401

Offset: 1

Author

Paul D. Hanna, Jun 23 2009

Keywords

• nonn

Examples

			E.g.f.: L(x) = x + 2*x^2/2! + 15*x^3/3! + 232*x^4/4! + 5905*x^5/5! +...
where exp(L(x)) = exp(x*exp(x*L'(x))) = e.g.f. of A161967:
exp(L(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 317*x^4/4! + 7596*x^5/5! +...
and exp(x*L'(x)) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! +...+ A156326(n)*x^n/n! +...
RELATED EXPRESSIONS.
E.g.f.: A(x) = 1 + 2*x + 15*x^2/2! + 232*x^3/3! + 5905*x^4/4! +...
where
A(x) = d/dx x*exp(x*A(x)) = exp(x*A(x)) * (1 + x*A(x) + x^2*A'(x)) with
exp(x*A(x)) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! + 1601497*x^6/6! + 92969920*x^7/7! +...+ A156326(n)*x^n/n! +...
		

Crossrefs

Cf. A161967 (exp), A156326.

Programs

PARI

{a(n)=local(L=x+x^2);for(i=1,n,L=x*exp(x*deriv(L)+O(x^n)));n!*polcoeff(L,n)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) = n * A156326(n-1), where the e.g.f. of A156326 satisfies: Sum_{n>=0} A156326(n)*x^n/n! = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ) = exp( Sum_{n>=1} n * a(n)*x^n/n! ). - Paul D. Hanna, Feb 21 2014

E.g.f. A(x), with offset=0, satisfies [Paul D. Hanna, Feb 15 2015]:

(1) A(x) = d/dx x*exp(x*A(x)).

(2) A(x) = exp(x*A(x)) * (1 + x*A(x) + x^2*A'(x)).

(3) exp(x*A(x)) = e.g.f. of A156326.