A356951 E.g.f. satisfies log(A(x)) = x^2/2 * (exp(x) - 1) * A(x).

1, 0, 0, 3, 6, 10, 285, 1911, 8848, 155016, 1931625, 17006275, 276807036, 4801114968, 65672925409, 1172625764415, 24657199159440, 460156401399376, 9560083801337793, 230955040794126915, 5393971086379904260, 131545127670380245920, 3587507216606547324321

Offset: 0

Author

Seiichi Manyama, Sep 06 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A052880, A355843, A356952.

Cf. A000272, A354000, A356752, A356949.

Programs

Mathematica

nmax = 22; A[_] = 1;
Do[A[x_] = Exp[x^2/2*(Exp[x] - 1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

PARI

a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*stirling(n-2*k, k, 2)/(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*(exp(x)-1))^k/k!)))

PARI

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

PARI

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).

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

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

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