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!)
A198631 Numerators of the rational sequence with e.g.f. 1/(1+exp(-x)). 19
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 2*A006519(n+1) when n is odd.

Also numerator of the value at the origin of the n-th derivative of the standard logistic function. - Enrique Pérez Herrero, Feb 15 2016

LINKS

Robert Israel, Table of n, a(n) for n = 0..550

Eric Weisstein's World of Mathematics, Sigmoid Function.

Wikipedia, Logistic Function.

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

Numerators of the Taylor series coefficients 4*(2^(n+1)-1)*B(n+1)/(n+1) for n>0 of 1 + 2 * tanh(x/2) (cf. A000182 and A089171). - Tom Copeland, Oct 19 2016

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

(Sage) # Alternatively:

def A198631_list(len):

    e, f, R, C = 2, 1, [], [1]+[0]*(len-1)

    for n in (1..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(numerator((e-1)*f*C[0]))

        f *= n; e <<= 1

    return R

print A198631_list(36) # Peter Luschny, Feb 21 2016

CROSSREFS

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

Cf. A000182, A089171.

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 by Peter Luschny, Jul 13 2012

Second comment corrected by Robert Israel, Feb 21 2016

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 10 22:40 EST 2016. Contains 279021 sequences.