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

1,2

Links

Table of n, a(n) for n=1..17.

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.

Example

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! +...

Prog

(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), ", "))

Crossrefs

Cf. A161967 (exp), A156326.

Sequence in context: A097628 A305111 A216860 * A294043 A326095 A212370

Adjacent sequences: A161965 A161966 A161967 * A161969 A161970 A161971

Keyword

nonn

Author

Paul D. Hanna, Jun 23 2009

Status

approved