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%I #11 Jan 07 2026 15:29:24
%S 1,1,1,-11,-47,-39,1681,10893,-13215,-851471,-5515679,34375397,
%T 887687857,3982645577,-85350572943,-1466457337859,-659043831871,
%U 270733024430305,3181606182917569,-24432689736388395,-1076204061663657839,-6834631528147762247,221729710998069153617
%N Expansion of e.g.f. exp(x * cos(2*x)).
%C As stated in the comment of A185951, A185951(n,0) = 0^n.
%F a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
%F a(n) = Sum_{k=0..n} (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit.
%t With[{nn=30},CoefficientList[Series[Exp[x Cos[2x]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jan 07 2026 *)
%o (PARI) a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
%o a(n) = sum(k=0, n, (2*I)^(n-k)*a185951(n, k));
%Y Cf. A009189, A381276.
%Y Cf. A185951.
%K sign
%O 0,4
%A _Seiichi Manyama_, Feb 18 2025