A356913 -id:A356913 - cp's OEIS frontend

A356753 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3/6 * A(x)).

1, 0, 0, 0, 4, 10, 40, 210, 3024, 25200, 225000, 2217600, 29974560, 400720320, 5558957040, 81340459200, 1344965825280, 23566775232000, 432681781459200, 8309927446329600, 170258024427580800, 3679448236206220800, 83235946152090547200, 1962840630226968307200

Offset: 0

Seiichi Manyama, Sep 03 2022

nonn

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

Cf. A351493, A356913.

Cf. A355842, A356752.

nmax = 23; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^3/6*A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

a(n) = n!*sum(k=0, n\4, (k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^3/6*log(1-x))^k/k!)))

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3/6*log(1-x))/(x^3/6*log(1-x))))

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^3/6 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^3/6 * log(1-x)) ).
E.g.f.: A(x) = LambertW(x^3/6 * log(1-x))/(x^3/6 * log(1-x)).

A356926 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^exp(x).

Original entry on oeis.org

1, 1, 2, 3, 10, 35, 121, 1092, 5216, 39321, 558643, 2433508, 48144944, 688652549, 2176310995, 145742587616, 1334993574032, 5551320939809, 799648465754835, 1049695714507276, 90069170433616208, 6281942689646504501, -53282051261767839293, 2356158301117802408472

Offset: 0

Seiichi Manyama, Sep 04 2022

sign

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

Cf. A191365, A356925.
Cf. A356912, A356913.
Cf. A002104, A177885.
Cf. A141209.

nmax = 23; A[_] = 1;
Do[A[x_] = ((1 - x)^(-Exp[x]))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-exp(x)*log(1-x))^k/k!)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-exp(x)*log(1-x)))))

my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*log(1-x)/lambertw(-exp(x)*log(1-x))))

E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-exp(x) * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-exp(x) * log(1-x)) ).
E.g.f.: A(x) = -exp(x) * log(1-x)/LambertW(-exp(x) * log(1-x)).

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A356753 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3/6 * A(x)).

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A356926 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^exp(x).

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