A219264 O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^2*x)^n/n! * exp(-n^2*x*A(n*x)).

1, 1, 8, 128, 3259, 120082, 6151625, 433404057, 42180568185, 5720993700540, 1088246094845838, 291276119631119408, 109983236494820652007, 58741463418913578672779, 44466318283501559718838424, 47771843216826858235974983400, 72930986725295232949801895385998

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..16.

Example

 O.g.f.: A(x) = 1 + x + 8*x^2 + 128*x^3 + 3259*x^4 + 120082*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(2*x)^2/2!*exp(-2^2*x*A(2*x)) + 3^6*x^3*A(3*x)^3/3!*exp(-3^2*x*A(3*x)) + 4^8*x^4*A(4*x)^4/4!*exp(-4^2*x*A(4*x)) + 5^10*x^5*A(5*x)^5/5!*exp(-5^2*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^(2*k)*x^k*subst(A, x, k*x)^k/k!*exp(-k^2*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

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

Sequence in context: A371299 A113135 A356972 * A349263 A188060 A104997

Adjacent sequences: A219261 A219262 A219263 * A219265 A219266 A219267

Keyword

nonn

Author

Paul D. Hanna, Nov 16 2012

Status

approved