login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 16
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)]

# 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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 2 15:01 EST 2016. Contains 278678 sequences.