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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. 3
1, 2, 15, 232, 5905, 220176, 11210479, 743759360, 62179950753, 6387468716800, 790466735915791, 115974842104378368, 19906425428056709425, 3952505003715017695232, 899034956269244372091375, 232282033898506324396343296, 67660142460130946247667502401 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 28 18:56 EST 2022. Contains 350657 sequences. (Running on oeis4.)