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 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

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