A355287 E.g.f. satisfies A(x) = 1/(1 - x)^(x^2 * A(x)).

1, 0, 0, 6, 12, 40, 1260, 8568, 62160, 1473120, 19111680, 232626240, 5403451680, 103176028800, 1822033204992, 45916616592000, 1129459815993600, 26346457488798720, 749439127417466880, 22165051763204582400, 640916967497214643200, 20787453048015928350720

Offset: 0

Author

Seiichi Manyama, Sep 03 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A353228, A356910.

Programs

Mathematica

nmax = 21; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^2*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\3, (k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!);

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^2*log(1-x))^k/k!)))

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2*log(1-x)))))

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2*log(1-x))/(x^2*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.

E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^2 * log(1-x))^k / k!.

E.g.f.: A(x) = exp( -LambertW(x^2 * log(1-x)) ).

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