A219343 O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^3*x)^n/n! * exp(-n^3*x*A(n*x)).

1, 1, 32, 3183, 650929, 226009218, 119298668857, 89086101638412, 89480710389500666, 116491795770107486363, 191172400354899371561288, 387419202671209086703674709, 956322827450633453264262285623, 2859815748552720894795327258080881, 10430012061189048036456303441601971435

Offset

0,3

Comments

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Links

Table of n, a(n) for n=0..14.

Example

 O.g.f.: A(x) = 1 + x + 32*x^2 + 3183*x^3 + 650929*x^4 + 226009218*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(2*x)^2/2!*exp(-2^3*x*A(2*x)) + 3^9*x^3*A(3*x)^3/3!*exp(-3^3*x*A(3*x)) + 4^12*x^4*A(4*x)^4/4!*exp(-4^3*x*A(4*x)) + 5^15*x^5*A(5*x)^5/5!*exp(-5^3*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Prog

 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^(3*k)*x^k*subst(A, x, k*x)^k/k!*exp(-k^3*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A219264, A219265, A218681, A218672, A219184, A219342, A217900.

Sequence in context: A135787 A016793 A215045 * A123802 A264057 A221659

Adjacent sequences: A219340 A219341 A219342 * A219344 A219345 A219346

Keyword

nonn

Author

Paul D. Hanna, Nov 18 2012

Status

approved