

A157779


Numerator of Bernoulli(n, 1/2).


8



1, 0, 1, 0, 7, 0, 31, 0, 127, 0, 2555, 0, 1414477, 0, 57337, 0, 118518239, 0, 5749691557, 0, 91546277357, 0, 1792042792463, 0, 1982765468311237, 0, 286994504449393, 0, 3187598676787461083, 0, 4625594554880206790555, 0, 16555640865486520478399, 0
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OFFSET

0,5


COMMENTS

Included for completeness, normally alternating zeros like this are omitted. A001896 is the official version of this sequence.
The sequence {a(n)/A141459(n)} gives the generalized Bernoulli numbers B[2,1] obtained from the generalized Stirling 2 triangle S3[2,1] = A154537. See the formula section.  Wolfdieter Lang, Apr 27 2017


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.


FORMULA

Let P(x)= Sum_{n>=0} x^(2*n+1)/(2*n+1)! then a(n) = numerator( n! [x^n] x/P(x) ).  Peter Luschny, Jul 05 2016
a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} ((1)^k / (k+1))*A154537(n, k)*k! = Sum_{k=0..n} ((1)^k/(k+1))*A145901(n, k). The denominators are in A141459. r(n) = B[2,1](n) = 2^n*B(n, 1/2) with the Bernoulli polynomials A196838/A196839 or A053382/A053383.  Wolfdieter Lang, Apr 27 2017


MATHEMATICA

Numerator[BernoulliB[Range[0, 40], 1/2]] (* Harvey P. Dale, May 04 2013 *)


PROG

(Sage)
def A157779_list(size):
f = x / sum(x^(n*2+1)/factorial(n*2+1) for n in (0..2*size))
t = taylor(f, x, 0, size)
return [(factorial(n)*s).numerator() for n, s in enumerate (t.list())]
print A157779_list(33) # Peter Luschny, Jul 05 2016
(PARI) a(n) = numerator(subst(bernpol(n, x), x, 1/2)); \\ Altug Alkan, Jul 05 2016


CROSSREFS

For denominators see A157780 and A141459.
Sequence in context: A282677 A280143 A280144 * A222322 A228762 A046273
Adjacent sequences: A157776 A157777 A157778 * A157780 A157781 A157782


KEYWORD

sign,frac


AUTHOR

N. J. A. Sloane, Nov 08 2009


STATUS

approved



