A226158 a(n) = 2*n*(2^n - 1)*zeta(1-n) where in the case n=0 the limit is understood, zeta(s) the Riemann zeta function.

0, -1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073, 0, -38227, 0, 929569, 0, -28820619, 0, 1109652905, 0, -51943281731, 0, 2905151042481, 0, -191329672483963, 0, 14655626154768697, 0, -1291885088448017715, 0, 129848163681107301953

Offset: 0

Author

Peter Luschny, Jun 28 2013

Keywords

• sign

Comments

Also known as the Genocchi numbers, apart from a(0) and a(1) same as A036968.

Consider the difference table of a(n), which is a variant of Seidel's Genocchi table A014781:

0 -1 -1 0 1 0 -3 0 17

-1 0 1 1 -1 -3 3 17 -17

1 1 0 -2 -2 6 14 -34 -138

0 -1 -2 0 8 8 -48 -104 448

-1 -1 2 8 0 -56 -56 552 1160

0 3 6 -8 -56 0 608 608 -8832

3 3 -14 -48 56 608 0 -9440 -9440

0 -17 -34 104 552 -608 -9440 0 198272

-17 -17 138 448 -1160 -8832 9440 198272 0

a(n) is an autosequence: its inverse binomial transform is the sequence signed (see A181722). The first column (inverse binomial transform) is 0, followed by -A036968. - Paul Curtz, Jul 22 2013

a(n+1) = p(0) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 1, ..., n+1. - Michael Somos, Apr 23 2014

Examples

			G.f. = - x - x^2 + x^4 - 3*x^6 + 17*x^8 - 155*x^10 + 2073*x^12 - 38227*x^14 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..550
• Peter Luschny, The Bernoulli Manifesto.

Crossrefs

Cf. A000367, A005439, A014781, A036968, A090648, A181722, A227577.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 50); [0] cat Coefficients(R!(Laplace( -2*x/(1+Exp(-x)) ))); // G. C. Greubel, Apr 22 2023

Maple

seq(n!*coeff(series(-2*x/(1+exp(-x)), x, 34), x, n), n=0..32);
# Second program:
A226158 := proc(n) local f; f := z -> Zeta(1-z)*2*z*(2^z-1);
if n=0 then limit(f(z), z=0) else f(n) fi end: seq(A226158(n), n=0..32);

Mathematica

a[0]=0; a[1]= -1; a[n_]:= n*EulerE[n-1, 0]; Table[a[n], {n,0,32}] (* Jean-François Alcover, Sep 12 2013 *)
(* Programs from Michael Somos, Apr 23 2014 *)
a[n_]:= If[n<1, 0, -n*EulerE[n-1, 1]];
a[n_]:= If[n<0, 0, 2*(1-2^n)*BernoulliB[n,1]]; (* End *)
Table[2*n*PolyLog[1-n, -1], {n,0,32}] (* Peter Luschny, Aug 17 2021 *)

PARI

my(x='x+O('x^40)); concat([0], Vec(serlaplace(-2*x/(1+exp(-x))))) \\ G. C. Greubel, Jan 19 2018

Sage

def A226158(n): return -2*n*zeta(1-n)*(1-2^n) if n != 0 else 0
[A226158(n) for n in (0..32)]
# Alternatively:
def A226158_list(len):
    e, f, R, C = 4, 1, [0], [1]+[0]*(len-1)
    for n in (2..len-1):
        for k in range(n, 0, -1):
            C[k] = -C[k-1] / (k+1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((2-e)*f*C[0])
        f *= n; e *= 2
    return R
print(A226158_list(34)) # Peter Luschny, Feb 22 2016

Formula

E.g.f.: -2*x/(1+exp(-x)).

a(2n) = -A000367(n)*A090648(n). - Paul Curtz, Jul 22 2013

E.g.f.: -2*x/(1+exp(-x))= -2 - 2*T(0), where T(k) = 4*k-1 + x/( 2 - x/( 4*k+1 + x/( 2 - x/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2013

G.f.: conjecture: -x/Q(0),where Q(k) = 1 - x*(k+1)/(1 + x*(k+1)/(1 - x*(k+1)/(1 + x*(k+1)/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2013

a(n) = 2*(1 - 2^n)*Bernoulli(n, 1). - Peter Luschny, Apr 16 2014

a(n) = -n*Euler(n - 1, 1). - Michael Somos, Apr 23 2014

a(n) = 2^n*(Bernoulli(n, 1/2) - Bernoulli(n, 1)). - Peter Luschny, Jul 10 2020

a(n) = 2*n*PolyLog[1 - n, -1] - Peter Luschny, Aug 17 2021