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%I #16 Mar 27 2022 02:15:14
%S 1,2,4,2,-32,-198,-416,2634,30720,107378,-605696,-10282094,-46020608,
%T 304968874,6121832448,29994597338,-279697555456,-5729595393310,
%U -26849178681344,401845799334690,7714801999937536,29062583111892506,-812705956979802112
%N Expansion of e.g.f. exp(2 * x * cos(x)).
%H Seiichi Manyama, <a href="/A352642/b352642.txt">Table of n, a(n) for n = 0..545</a>
%F a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
%t With[{m = 22}, Range[0, m]! * CoefficientList[Series[Exp[2*x*Cos[x]], {x, 0, m}], x]] (* _Amiram Eldar_, Mar 26 2022 *)
%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*x*cos(x))))
%o (PARI) a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*binomial(n-1, 2*k)*a(n-2*k-1)));
%Y Cf. A009189, A352643.
%K sign
%O 0,2
%A _Seiichi Manyama_, Mar 25 2022
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