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%I #41 Feb 07 2019 11:51:03
%S 1,-1,1,-17,31,-691,5461,-929569,3202291,-221930581,4722116521,
%T -56963745931,14717667114151,-2093660879252671,86125672563201181,
%U -129848163681107301953,868320396104950823611,-209390615747646519456961,14129659550745551130667441
%N Numerators of series coefficients of 1/(1 + cosh(sqrt(x))).
%C Unsigned version is equal to A002425 up to n=11, but differs beyond that point.
%C Unsigned version: numerators of series coefficients of 1/(1 + cos(sqrt(x))); see Mathematica. - _Clark Kimberling_, Dec 06 2016
%H N. J. A. Sloane, <a href="/A089171/b089171.txt">Table of n, a(n) for n = 0..299</a>
%F a(n) = numerator(c(n+1)) where c(n)=(2^(2*n)-1)*B(2*n)/(2*n)!, B(k) denotes the k-th Bernoulli number. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004
%F Numerators of expansion of cosec(x)-cot(x) = 1/2*x+1/4*x^3/3!+1/2*x^5/5!+17/8*x^7/7!+31/2*x^9/9!+... - _Ralf Stephan_, Dec 21 2004 (Comment was applied to wrong entry, corrected by Alessandro Musesti (musesti(AT)gmail.com), Nov 02 2007)
%F E.g.f.: 1/sin(x)-cot(x). - _Sergei N. Gladkovskii_, Nov 22 2011
%F E.g.f.: x/G(0); G(k)=4k+2-x^2/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 22 2011
%F E.g.f.: (1+x/(x-2*Q(0))/2; Q(k)=8k+2+x/(1+(2k+1)*(2k+2)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Nov 22 2011
%F E.g.f.: x/(x+Q(0)); Q(k)=x+(x^2)/((4*k+1)*(4*k+2)-(4*k+1)*(4*k+2)/(1+(4*k+3)*(4*k+4)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 22 2011
%F E.g.f.: T(0)/2, where T(k) = 1 - x^2/(x^2 - (4*k+2)*(4*k+6)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 12 2013
%F Aerated, these are the numerators of the Taylor series coefficients of 2 * tanh(x/2) (cf. A000182 and A198631). - _Tom Copeland_, Oct 19 2016
%p with(numtheory): c := n->(2^(2*n)-1)*bernoulli(2*n)/(2*n)!; seq(numer(c(n)),n=1..20); # C. Ronaldo
%t Numerator[CoefficientList[Series[1/(1+Cosh[Sqrt[x]]), {x, 0, 24}], x]]
%t Numerator[CoefficientList[Series[1/(1+Cos[Sqrt[x]]), {x, 0, 30}], x]]
%t (* unsigned version, _Clark Kimberling_, Dec 06 2016 *)
%Y Cf. A002425, A050970, A002430, A046990.
%Y Cf. A000182, A198631, A279009, A279010.
%Y Cf. A276592, A279370.
%K sign,frac
%O 0,4
%A _Wouter Meeussen_, Dec 07 2003
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