A305922 Expansion of e.g.f. log(1 + 2*x/(exp(x) + 1)).

0, 1, -2, 5, -20, 109, -738, 5991, -56760, 614601, -7486670, 101330635, -1508641140, 24503026989, -431137315434, 8169513007215, -165859346028656, 3591802533860497, -82644488286784326, 2013441061219406739, -51777972823724776620, 1401611202556240950645, -39838169568923591411810

Offset

0,3

Comments

Logarithmic transform of A036968.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..430

N. J. A. Sloane, Transforms

Wikipedia, Genocchi number

Index entries for sequences related to Bernoulli numbers

Example

E.g.f.: A(x) = x - 2*x^2/2! + 5*x^3/3! - 20*x^4/4! + 109*x^5/5! - 738*x^6/6! + ...

Maple

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(a(j)*j*
      t(n-j)*binomial(n, j), j=1..n-1)/n))(i-> i*euler(i-1, 0))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 04 2018

Mathematica

nmax = 22; CoefficientList[Series[Log[1 + 2 x/(Exp[x] + 1)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = n EulerE[n - 1, 0] - Sum[k Binomial[n, k] (n - k) EulerE[n - k - 1, 0] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]
a[n_] := a[n] = 2 (1 - 2^n) BernoulliB[n] - Sum[k Binomial[n, k] 2 (1 - 2^(n - k)) BernoulliB[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]

Crossrefs

Cf. A036968, A296837, A296838, A305711.

Sequence in context: A170946 A296727 A201224 * A019536 A129949 A127065

Adjacent sequences: A305919 A305920 A305921 * A305923 A305924 A305925

Keyword

sign

Author

Ilya Gutkovskiy, Jun 14 2018

Status

approved