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A009628 Expansion of e.g.f.: sinh(x)/(1+x).

%I A009628 #51 Mar 19 2023 18:00:52
%S A009628 0,1,-2,7,-28,141,-846,5923,-47384,426457,-4264570,46910271,
%T A009628 -562923252,7318002277,-102452031878,1536780478171,-24588487650736,
%U A009628 418004290062513,-7524077221125234,142957467201379447,-2859149344027588940,60042136224579367741
%N A009628 Expansion of e.g.f.: sinh(x)/(1+x).
%C A009628 (-1)^n*(A000166 + A000522)/2 = A009179, (-1)^n*(A000166-A000522)/2 = this_sequence.
%H A009628 Seiichi Manyama, <a href="/A009628/b009628.txt">Table of n, a(n) for n = 0..449</a>
%F A009628 a(n) = (-1)^(n+1)*floor(n!*sinh(1)), n>=1. - _Vladeta Jovovic_, Aug 10 2002
%F A009628 Let u(1) = 1, u(n) = n*u(n-1) + n (mod 2); then for n>0, a(n) = (-1)^(n+1)*u(n). - _Benoit Cloitre_, Jan 12 2003
%F A009628 Unsigned sequence satisfies a(n) = n*a(n-1)+a(n-2)-(n-2)*a(n-3), with E.g.f. sinh(z)/(1-z). - Mario Catalani (mario.catalani(AT)unito.it), Feb 08 2003
%F A009628 a(n) = (-1)^(n+1) * n! * Sum_{k=1..floor((n+1)/2)} 1/(2*k-1)!.
%F A009628 a(n) = -n*a(n-1) + n (mod 2). - _Seiichi Manyama_, Sep 09 2016
%F A009628 a(n) = (-1)^n*(exp(-1)*Gamma(1+n,-1) - exp(1)*Gamma(1+n,1))/2. - _Peter Luschny_, Dec 18 2017
%p A009628 G(x):= sinh(x)/(1+x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # _Zerinvary Lajos_, Apr 03 2009
%t A009628 a[n_] := (-1)^n (Exp[-1] Gamma[1 + n, -1] - Exp[1] Gamma[1 + n, 1])/2;
%t A009628 Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Dec 18 2017 *)
%t A009628 With[{nn=30},CoefficientList[Series[Sinh[x]/(1+x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 19 2023 *)
%o A009628 (PARI) a(n) = n!*polcoeff((sinh(x)/(1+x) + x * O(x^n)), n) \\ _Charles R Greathouse IV_, Sep 09 2016
%o A009628 (PARI) x='x+O('x^99); concat([0], Vec(serlaplace(sinh(x)/(1+x)))) \\ _Altug Alkan_, Dec 18 2017
%o A009628 (Ruby)
%o A009628 def A009628(n)
%o A009628   a = 0
%o A009628   (0..n).map{|i| a = -i * a + i % 2}
%o A009628 end # _Seiichi Manyama_, Sep 09 2016
%Y A009628 Cf. A000166, A000522, A009179, A051397, A087208, A186763.
%K A009628 sign,easy
%O A009628 0,3
%A A009628 _R. H. Hardin_
%E A009628 Extended with signs by _Olivier Gérard_, Mar 15 1997
%E A009628 Definition clarified by _Harvey P. Dale_, Mar 19 2023