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%I #31 Sep 08 2022 08:44:37
%S 1,0,-1,3,-2,-20,74,98,-1532,960,41324,-105732,-1595912,7998640,
%T 85401224,-705417112,-6026865392,76352075520,537223559024,
%U -10130428275792,-58185728893472,1628892022801600,7352490891960224,-313251680404802272
%N Expansion of e.g.f. cos(log(1+tanh(x))).
%C Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times Sum_{k=0..n} binomial(n,k)*k^q, then A009022(x) equals (-1)^(x+1) times the imaginary part of Sum_{k=0..x-1} T(x,k)*i^k, where i is the imaginary unit. See Mathematica code below. - _John M. Campbell_, Nov 17 2011
%H Vincenzo Librandi, <a href="/A009022/b009022.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{m=0..n/2} (-1)^(m)*Sum_{r=2*m..n} (Stirling1(r,2*m)* Sum_{k=r..n} binomial(k-1,r-1)*k!*2^(n-k)*Stirling2(n,k)*(-1)^(r+k))/r!), n > 0, a(0)=1. - _Vladimir Kruchinin_, Jun 21 2011
%t Join[{1}, Cos[Log[1 + Tanh[x]]];
%t poly[q_] := 2^(q - n)/n FunctionExpand[Sum[Binomial[n, k] k^q, {k, 0, n}]]; T[q_, r_] := First[Take[CoefficientList[poly[q], n], {r + 1, r + 1}]]; Table[Im[Sum[T[x, k] I^k, {k, 0, x - 1}]] (-1)^(x + 1), {x, 1, 23}]] (* _John M. Campbell_, Nov 17 2011 *)
%t With[{nn = 30}, Take[CoefficientList[Series[Cos[Log[1 + Tanh[x]]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 1}]] (* _Vincenzo Librandi_, Feb 09 2014 *)
%o (Maxima)
%o a(n):=if n=0 then 1 else sum((-1)^(m)*sum((stirling1(r,2*m)*sum(binomial(k-1,r-1)*k!*2^(n-k)*stirling2(n,k)*(-1)^(r+k),k,r,n))/r!,r,2*m,n),m,0,n/2); /* _Vladimir Kruchinin_, Jun 21 2011 */
%o (PARI) x='x+O('x^30); Vec(serlaplace(cos(log(1+tanh(x))))) \\ _G. C. Greubel_, Jul 22 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cos(Log(1+Tanh(x))))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 22 2018
%K sign,easy
%O 0,4
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_ Mar 15 1997
%E Adapted Campbell's Mathematica program for offset by _Vincenzo Librandi_, Feb 09 2014
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