A357009 E.g.f. satisfies log(A(x)) = (exp(x) - 1)^2 * A(x).

1, 0, 2, 6, 50, 390, 4322, 53046, 782210, 12920550, 241747682, 5000171286, 113961184130, 2830240421190, 76196913418082, 2209152734071926, 68655746019566210, 2276606079902438310, 80244521295497399522, 2995966456305973559766, 118119901491333724203650

Offset: 0

Author

Seiichi Manyama, Sep 09 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A052880, A357010.

Cf. A052859, A357024.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[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) = sum(k=0, n\2, (2*k)!*(k+1)^(k-1)*stirling(n, 2*k, 2)/k!);

PARI

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

PARI

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

PARI

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

Formula

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

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

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

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

a(n) ~ sqrt(1 + exp(1/2)) * 2^n * n^(n-1) / (exp(n-1) * (2*log(1 + exp(1/2)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Sep 27 2023