A355308 Expansion of e.g.f. -LambertW(x^3/6 * (1 - exp(x))).

0, 0, 0, 0, 4, 10, 20, 35, 1176, 10164, 58920, 277365, 4472380, 69189406, 772011604, 6861855455, 95279504880, 1819310613800, 30768119885136, 430200439251369, 6770486332450740, 139958614722287410, 3033142442978720380, 58782387380290683571, 1138026666874389737544

Offset

0,5

Links

Table of n, a(n) for n=0..24.

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

Formula

a(n) = n! * Sum_{k=1..floor(n/4)} k^(k-1) * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

a(n) ~ sqrt(exp(r)*(3+r)/6 - 1/2) * n^(n-1) / (exp(n - 1/2) * r^(n - 3/2)), where r = 1.055946740386774512329072722491942575326676434456... is the root of the equation exp(r+1) - exp(1) = 6/r^3. - Vaclav Kotesovec, Jan 31 2026

Prog

(PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3/6*(1-exp(x))))))
(PARI) a(n) = n!*sum(k=1, n\4, k^(k-1)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Crossrefs

Cf. A048802, A355181, A357267.

Cf. A353999, A355180.

Sequence in context: A354001 A368174 A353999 * A356952 A356963 A375716

Adjacent sequences: A355305 A355306 A355307 * A355309 A355310 A355311

Keyword

nonn

Author

Seiichi Manyama, Sep 24 2022

Status

approved