A356752 E.g.f. satisfies A(x) = 1/(1 - x)^(x^2/2 * A(x)).

1, 0, 0, 3, 6, 20, 360, 2394, 17220, 260280, 3076920, 35980560, 595686960, 9760411440, 159321570408, 3093987619800, 63314740616400, 1318245318411840, 30240056863978560, 736919729169603840, 18522487833889334400, 495842871278901363840, 14014346231616983128800

Offset: 0

Author

Seiichi Manyama, Sep 03 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A351492, A356912.

Cf. A355842, A356753.

Programs

Mathematica

nmax = 22; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^2/2*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))/(2^k*(n-2*k)!));

PARI

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

PARI

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

PARI

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

Formula

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

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

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

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