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%I #55 Aug 04 2024 03:47:11
%S 0,1,-1,1,0,-10,90,-730,6160,-55900,549900,-5864300,67610400,
%T -839594600,11186357000,-159300557000,2416003824000,-38894192662000,
%U 662595375078000,-11911522255750000,225382826562400000
%N Expansion of e.g.f. sin(log(1+x)).
%F a(n) = Sum_{k=0..n-1} (-1)^k*T(n-1, k)*cos(Pi*(n-k-1)/2); T(n, k) = abs(A008276(n, k)). - _Paul Barry_, Apr 18 2005
%F abs(a(n)) = abs(Re(Product_{k=1..n-1} (k+I))) with I^2 = -1. - _Yalcin Aktar_, Jul 02 2005
%F a(n+2) = -(2n+1)*a(n+1)-(n^2+1)*a(n), a(0)=0, a(1)=1. - Remy Lachaud (pacifik31(AT)aol.com), Dec 25 2005
%F a(n) = Sum_{k=0..n/2} Stirling1(n,2k+1)*(-1)^k. - _Vladimir Kruchinin_, Aug 03 2010
%F a(n) = Im(gamma(i+1)/gamma(i+1-n)). The real part is A003703. - _Colin Beveridge_, Jul 30 2024
%t CoefficientList[Series[Sin[Log[1+x]], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 24 2015 *)
%t FullSimplify[Table[-((-1)^n*(Gamma[1 + I]*Gamma[-I + n] + Gamma[1 - I]*Gamma[I + n])*Sinh[Pi]) / (2*Pi), {n, 0, 20}]] (* _Vaclav Kotesovec_, Jan 24 2015 *)
%t Table[-(-1)^n Re[Pochhammer[1+I, n-1]], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 13 2016 *)
%o (Maxima) sum(stirling1(n,2*k+1)*(-1)^(k),k,0,n/2) /* _Vladimir Kruchinin_, Aug 03 2010 */
%o (Python)
%o from sympy.functions.combinatorial.numbers import stirling
%o def A009454(n): return sum(stirling(n,(k<<1)+1,kind=1,signed=True)*(-1 if k&1 else 1) for k in range(n+1>>1)) # _Chai Wah Wu_, Feb 22 2024
%Y Cf. A003703, A242652, A231530.
%K sign,easy
%O 0,6
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
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