A356913 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3/6).

1, 0, 0, 0, 4, 10, 40, 210, 784, 5040, 40200, 369600, 5285280, 72072000, 1006889520, 14760345600, 210510263040, 3131345817600, 49229619129600, 818940523564800, 15054020163619200, 301204611031564800, 6455999452413772800, 146587705490513548800

Offset: 0

Author

Seiichi Manyama, Sep 03 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A356905, A356912.

Cf. A351493, A356753.

Programs

Mathematica

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

PARI

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

PARI

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!)))

PARI

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

PARI

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

Formula

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) = -x^3/6 * log(1-x)/LambertW(-x^3/6 * log(1-x)).