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A226158 a(n) = zeta(1-n)*2*n*(2^n-1) where in the case n=0 the limit is understood, zeta(s) the Riemann zeta function. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..550

Peter Luschny, The Bernoulli Manifesto.

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

EXAMPLE

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

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-Fran├žois Alcover, Sep 12 2013 *)

a[ n_] := If[ n < 1, 0, -n EulerE[n - 1, 1]]; (* Michael Somos, Apr 23 2014 *)

a[ n_] := If[ n < 0, 0, 2 (1 - 2^n) BernoulliB[n, 1]]; (* Michael Somos, Apr 23 2014 *)

PROG

(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)]

CROSSREFS

Cf. A036968, A227577, A005439.

Sequence in context: A143779 A240244 A036968 * A024040 A009759 A127187

Adjacent sequences:  A226155 A226156 A226157 * A226159 A226160 A226161

KEYWORD

sign

AUTHOR

Peter Luschny, Jun 28 2013

STATUS

approved

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Last modified August 27 15:19 EDT 2015. Contains 261092 sequences.