A372157 E.g.f. A(x) satisfies A(x) = exp( 3 * x * (1 + x * A(x)^(1/3)) ).

1, 3, 15, 99, 837, 8583, 104229, 1463031, 23362089, 418489227, 8318989089, 181823016147, 4335947796717, 112073459278095, 3122026815194205, 93267116190237807, 2974988801559127761, 100932108044810678547, 3629658464478098931897, 137928467585817206673291

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

E.g.f.: A(x) = exp( 3*x - 3*LambertW(-x^2 * exp(x)) ).

If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(s*k,n-k)/k!.

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3*x-3*lambertw(-x^2*exp(x)))))
(PARI) a(n, r=3, s=1, t=0, u=1) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(s*k, n-k)/k!);

Crossrefs

Cf. A125500, A372156.

Sequence in context: A255806 A226515 A135883 * A147664 A219779 A303651

Adjacent sequences: A372154 A372155 A372156 * A372158 A372159 A372160

Keyword

nonn

Author

Seiichi Manyama, Apr 20 2024

Status

approved