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

%I A198631 #85 Feb 16 2025 08:33:15
%S A198631 1,1,0,-1,0,1,0,-17,0,31,0,-691,0,5461,0,-929569,0,3202291,0,
%T A198631 -221930581,0,4722116521,0,-968383680827,0,14717667114151,0,
%U A198631 -2093660879252671,0,86125672563201181,0,-129848163681107301953,0,868320396104950823611,0
%N A198631 Numerators of the rational sequence with e.g.f. 1/(1+exp(-x)).
%C A198631 Numerators of the row sums of the Euler triangle A060096/A060097.
%C A198631 The corresponding denominator sequence looks like 2*A006519(n+1) when n is odd.
%C A198631 Also numerator of the value at the origin of the n-th derivative of the standard logistic function. - _Enrique Pérez Herrero_, Feb 15 2016
%H A198631 Robert Israel, <a href="/A198631/b198631.txt">Table of n, a(n) for n = 0..550</a>
%H A198631 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SigmoidFunction.html">Sigmoid Function</a>.
%H A198631 Wikipedia, <a href="https://en.wikipedia.org/wiki/Logistic_function">Logistic Function</a>.
%F A198631 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).
%F A198631 E.g.f.: 2/(1+exp(-x)) (see a comment in  A060096).
%F A198631 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
%F A198631 a(n) = numerator(euler(n,1)/(2^n-1)) for n > 0. - _Peter Luschny_, Jul 14 2013
%F A198631 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
%F A198631 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
%F A198631 a(n) = -2*zeta(-n)*A335956(n+1). - _Peter Luschny_, Jul 21 2020
%F A198631 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
%e A198631 The rational sequence r(n) = a(n) / A006519(n+1) starts:
%e A198631 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, ...
%p A198631 seq(denom(euler(i,x))*euler(i,1),i=0..33); # _Peter Luschny_, Jun 16 2012
%t A198631 Join[{1},Table[Numerator[EulerE[n,1]/(2^n-1)], {n, 34}]] (* _Peter Luschny_, Jul 14 2013 *)
%o A198631 (Sage)
%o A198631 def A198631_list(n):
%o A198631     x = var('x')
%o A198631     s = (1/(1+exp(-x))).series(x,n+2)
%o A198631     return [(factorial(i)*s.coefficient(x,i)).numerator() for i in (0..n)]
%o A198631 A198631_list(34) # _Peter Luschny_, Jul 12 2012
%o A198631 (Sage) # Alternatively:
%o A198631 def A198631_list(len):
%o A198631     e, f, R, C = 2, 1, [], [1]+[0]*(len-1)
%o A198631     for n in (1..len-1):
%o A198631         for k in range(n, 0, -1):
%o A198631             C[k] = -C[k-1] / (k+1)
%o A198631         C[0] = -sum(C[k] for k in (1..n))
%o A198631         R.append(numerator((e-1)*f*C[0]))
%o A198631         f *= n; e <<= 1
%o A198631     return R
%o A198631 print(A198631_list(36)) # _Peter Luschny_, Feb 21 2016
%Y A198631 Cf. A000182, A060096, A060097, A006519, A002425, A089171, A090681.
%K A198631 sign,easy,frac
%O A198631 0,8
%A A198631 _Wolfdieter Lang_, Oct 31 2011
%E A198631 New name, a simpler standalone definition by _Peter Luschny_, Jul 13 2012
%E A198631 Second comment corrected by _Robert Israel_, Feb 21 2016