A357027 E.g.f. satisfies A(x) = 1/(1 - x)^(log(1 - x)^2 * A(x)).

1, 0, 0, 6, 36, 210, 2430, 32424, 426552, 6575304, 118916640, 2328078456, 49421111256, 1153979875152, 29201577206256, 791744021665344, 22988121190902720, 712541051083100160, 23447653175729566080, 816434611464004145280, 30009023179153182132480

Offset: 0

Author

Seiichi Manyama, Sep 09 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A052813, A357026.

Programs

Mathematica

m = 21; (* number of terms *)
A[_] = 0;
Do[A[x_] = 1/(1 - x)^(Log[1 - x]^2*A[x]) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)

PARI

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

PARI

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

PARI

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

PARI

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

Formula

E.g.f. satisfies log(A(x)) = -log(1 - x)^3 * A(x).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (k+1)^(k-1) * |Stirling1(n,3*k)|/k!.

E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-log(1 - x))^(3*k) / k!.

E.g.f.: A(x) = exp( -LambertW(log(1-x)^3) ).

E.g.f.: A(x) = LambertW(log(1 - x)^3)/log(1 - x)^3.