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

0,3

Links

Table of n, a(n) for n=0..20.

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

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.

Prog

(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))

Crossrefs

Cf. A052880, A357010.

Cf. A052859, A357024.

Sequence in context: A080310 A103990 A079835 * A177454 A357086 A052332

Adjacent sequences: A357006 A357007 A357008 * A357010 A357011 A357012

Keyword

nonn

Author

Seiichi Manyama, Sep 09 2022

Status

approved