A198631 Numerators of the rational sequence with e.g.f. 1/(1+exp(-x)).

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

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

a(n) = -2*zeta(-n)*A335956(n+1). - Peter Luschny, Jul 21 2020

Conjecture: r(n) = Sum_{k=0..n} A001147(k) * A039755(n, k) * (-1)^k / (k+1) where r(n) = a(n) / A006519(n+1) = (n!) * ([x^n] (2 / (1 + exp(-x)))), for n >= 0. - Werner Schulte, Feb 16 2024

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):
    x = var('x')
    s = (1/(1+exp(-x))).series(x, n+2)
    return [(factorial(i)*s.coefficient(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. A000182, A060096, A060097, A006519, A002425, A089171, A090681.

Sequence in context: A341689 A243776 A338801 * A185685 A144692 A335264

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