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%I #24 Apr 16 2026 09:59:39
%S 0,1,-10,256,-13840,1282816,-181649920,36478744576,-9861412096000,
%T 3452931391553536,-1520176422094766080,821908023102140317696,
%U -535369559186849435484160,413508899951810581475885056,-373681769282729298809313034240,390605070751731814007608616943616
%N Expansion of e.g.f. log(1 + sin(x)^2)/2 (even-indexed coefficients).
%F A(0,k) = 1 and A(n,k) = -(k*(k+1) * A(n-1,k+2) + k^2 * A(n-1,k)) for n > 0. a(0) = 0 and a(n) = A(n-1,2). - _Seiichi Manyama_, Apr 15 2026
%e log(1 + sin(x)^2)/2 = x^2/2! - 10*x^4/4! + 256*x^6/6! - 13840*x^8/8! + 1282816*x^10/10! - ...
%t With[{nn=30},Take[CoefficientList[Series[Log[1+Sin[x]^2]/2,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, May 21 2014 *)
%Y Cf. A009344, A012494.
%K sign
%O 0,3
%A _R. H. Hardin_
%E Extended with signs Mar 1997
%E Prior Mathematica program replaced by _Harvey P. Dale_, May 21 2014