A357240 Expansion of e.g.f. 2 * (exp(x) - 1) / (exp(exp(x) - 1) + 1).

0, 1, 0, -2, -5, -4, 32, 225, 794, 190, -22291, -200298, -920244, 924223, 65848880, 716920754, 3831260555, -13147083976, -575844827780, -7162425813919, -40755845041730, 320194436283162, 11810647258173653, 161108090793013130, 896865861205240824, -14305712791762925929, -487306962045115504436

Offset

0,4

Comments

Stirling transform of the Genocchi numbers (of first kind, A036968).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..494

Formula

a(n) = 2 * Sum_{k=0..n} Stirling2(n,k) * (1 - 2^k) * Bernoulli(k).

a(n) ~ Pi^(3/2) * 2^(n + 7/2) * n^(n + 1/2) * (cos(n*arctan(2*arctan(Pi)/log(1 + Pi^2))) * (Pi*log(1 + Pi^2) + 2*arctan(Pi)) + (log(1 + Pi^2) - 2*Pi*arctan(Pi)) * sin(n*arctan(2*arctan(Pi)/log(1 + Pi^2)))) / ((1 + Pi^2) * exp(n) * (4*arctan(Pi)^2 + log(1 + Pi^2)^2)^(n/2 + 1)). - Vaclav Kotesovec, Oct 04 2022

Maple

b:= proc(n, m) option remember; `if`(n=0, `if`(m=0, 0,
      m*euler(m-1, 0)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..26);  # Alois P. Heinz, Jun 23 2023

Mathematica

nmax = 26; CoefficientList[Series[2 (Exp[x] - 1)/(Exp[Exp[x] - 1] + 1), {x, 0, nmax}], x] Range[0, nmax]!
Table[2 Sum[StirlingS2[n, k] (1 - 2^k) BernoulliB[k], {k, 0, n}], {n, 0, 26}]

Prog

(PARI) a(n) = 2*sum(k=0, n, stirling(n, k, 2)*(1-2^k)*bernfrac(k)); \\ Michel Marcus, Sep 20 2022

Crossrefs

Cf. A001469, A003149, A036968, A059371.

Sequence in context: A128202 A319770 A229789 * A307131 A210419 A206484

Adjacent sequences: A357237 A357238 A357239 * A357241 A357242 A357243

Keyword

sign

Author

Ilya Gutkovskiy, Sep 19 2022

Status

approved