A355179 -id:A355179 - cp's OEIS frontend

A355180 Expansion of e.g.f. -LambertW(x^3 * (1 - exp(x)))/6.

0, 0, 0, 0, 4, 10, 20, 35, 6776, 60564, 352920, 1663365, 126625180, 2361079006, 27334747804, 245495250455, 11174333090480, 328952158255400, 6245314009946736, 90576650639967369, 3209305759254634740, 122557203047084965810, 3365068665450300234580

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

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

Cf. A048802, A355179, A357267.

my(N=20, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3*(1-exp(x)))/6)))

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

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

A355874 Expansion of e.g.f. -LambertW(x^2 * log(1-x))/2.

Original entry on oeis.org

0, 0, 0, 3, 6, 20, 450, 3024, 21840, 449280, 5690160, 68579280, 1491462720, 27798076800, 485405784864, 11821894207200, 285057334598400, 6578025489584640, 183420564173141760, 5342163886869062400, 152988752430721267200, 4897735504358795965440

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

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

Cf. A052807, A355993, A357265.

Cf. A355179.

With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^2 * Log[1 - x]]/2, {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)

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

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

a(n) = (n!/2) * Sum_{k=1..floor(n/3)} k^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.

cp's OEIS Frontend

A355180 Expansion of e.g.f. -LambertW(x^3 * (1 - exp(x)))/6.

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A355874 Expansion of e.g.f. -LambertW(x^2 * log(1-x))/2.

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