A357424 E.g.f. satisfies A(x) * exp(A(x)) = exp(x * exp(A(x))) - 1.

0, 1, 1, 4, 21, 156, 1470, 16843, 227367, 3533974, 62163477, 1220852524, 26480355110, 628693388909, 16216901961481, 451609382251836, 13504072800481613, 431544662700594212, 14677503631085378170, 529370720888418692643, 20180856622352239827687

Offset: 0

Author

Seiichi Manyama, Sep 27 2022

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A052888, A357346, A357347, A357348.

Cf. A349588.

Programs

Mathematica

Join[{0,1},Table[Sum[(n-k)^(k-1) * StirlingS2[n,k], {k,1,n}], {n,2,20}]] (* Vaclav Kotesovec, Nov 14 2022 *)

PARI

a(n) = sum(k=1, n, (n-k)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (n-k)^(k-1) * Stirling2(n,k).

a(n) ~ n^(n-1) * (1 + exp(s)*s)^(n + 1/2) / (sqrt(exp(s)*(1 + s + s^2) - 1) * exp(n) * (1 + s)^(n - 1/2)), where s = 1.104072744884035178291292242554731... is the root of the equation 1 + s = (exp(-s) + s) * log(1 + exp(s)*s). - Vaclav Kotesovec, Nov 14 2022

E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(x)) ). - Seiichi Manyama, Sep 09 2024