A355180 -id:A355180 - cp's OEIS frontend

A355179 Expansion of e.g.f. -LambertW(x^2 * (1 - exp(x)))/2.

0, 0, 0, 3, 6, 10, 375, 2541, 11788, 317556, 4238685, 37921015, 909616026, 18283276518, 261259582675, 6360432558585, 164704011195480, 3332419310132776, 88606184592031353, 2713050497589230763, 71412977041725823750, 2144089948615678382970

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

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

Cf. A048802, A355180, A357267.

With[{nn=30},CoefficientList[Series[(-LambertW[x^2 (1-Exp[x])])/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 07 2025 *)

my(N=20, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(-lambertw(x^2*(1-exp(x)))/2)))

a(n) = n!*sum(k=1, n\3, k^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!)/2;

a(n) = (n!/2) * Sum_{k=1..floor(n/3)} k^(k-1) * Stirling2(n-2*k,k)/(n-2*k)!.

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

Original entry on oeis.org

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

Seiichi Manyama, Sep 24 2022

nonn

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

Cf. A048802, A355181, A357267.

Cf. A353999, A355180.

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3/6*(1-exp(x))))))

a(n) = n!*sum(k=1, n\4, k^(k-1)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

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

cp's OEIS Frontend

A355179 Expansion of e.g.f. -LambertW(x^2 * (1 - exp(x)))/2.

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

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