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A198631 Numerators of the rational sequence with e.g.f. 1/(1+exp(-x)). 15
1, 1, 0, -1, 0, 1, 0, -17, 0, 31, 0, -691, 0, 5461, 0, -929569, 0, 3202291, 0, -221930581, 0, 4722116521, 0, -968383680827, 0, 14717667114151, 0, -2093660879252671, 0, 86125672563201181, 0, -129848163681107301953, 0, 868320396104950823611, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Numerators of the row sums of the Euler triangle A060096/A060097.

The corresponding denominator sequence looks like A006519(n+1).

LINKS

Table of n, a(n) for n=0..34.

FORMULA

a(n) = numerator(sum(E(n,m),m=0..n)), n>=0, with the Euler triangle E(n,m)=A060096(n,m)/A060097(n,m).

E.g.f.: 2/(1+exp(-x)) (see a comment in  A060096).

r(n) := sum(E(n,m),m=0..n) = ((-1)^n)*sum(((-1)^m)*m!*S2(n,m)/2^m, m=0..n), n>=0, where S2 are the Stirling numbers of the second kind A048993. From the e.g.f. with y=exp(-x), dx=-y*dy, putting y=1 at the end. - Wolfdieter Lang, Nov 03 2011

a(n) = numerator(euler(n,1)/(2^n-1)) for n > 0. - Peter Luschny, Jul 14 2013

a(n) = numerator(2*(2^n-1)*B(n,1)/n) for n > 0, B(n,x) the Bernoulli polynomials. - Peter Luschny, May 24 2014

EXAMPLE

The rational sequence r(n)=a(n)/A006519(n+1) starts 1,1/2,0,-1/4,0,1/2,0,-17/8,0,31/2,0,-691/4,0,

  5461/2,0,-929569/16,0, 3202291/2,0,-221930581/4, 0,4722116521/2,0,-968383680827/8,0,14717667114151/2,0,

  -2093660879252671/4,...

MAPLE

seq(denom(euler(i, x))*euler(i, 1), i=0..33); # Peter Luschny, Jun 16 2012

MATHEMATICA

Join[{1}, Table[Numerator[EulerE[n, 1]/(2^n-1)], {n, 34}]] (* Peter Luschny, Jul 14 2013 *)

PROG

(Sage)

def A198631_list(n) :

    s = (1/(1+exp(-x))).series(x, n+2)

    return [(factorial(i)*s.coeff(x, i)).numerator() for i in (0..n)]

A198631_list(34) # Peter Luschny, Jul 12 2012

CROSSREFS

Cf. A060096, A060097, A006519, A002425, A089171, A090681.

Sequence in context: A059933 A002488 A243776 * A185685 A144692 A241027

Adjacent sequences:  A198628 A198629 A198630 * A198632 A198633 A198634

KEYWORD

sign,easy,frac

AUTHOR

Wolfdieter Lang, Oct 31 2011

EXTENSIONS

New name, a simpler standalone definition. Peter Luschny, Jul 13 2012.

STATUS

approved

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Last modified November 24 16:46 EST 2014. Contains 249899 sequences.