A352639 Expansion of e.g.f. exp(2*sin(x)).
1, 2, 4, 6, 0, -46, -192, -266, 1792, 14114, 34816, -171930, -2027520, -6522382, 34750464, 496296022, 1748500480, -12731696062, -186550845440, -617309234490, 7292215885824, 99199654760978, 248883934396416, -5836506132182090, -69729013345550336
Offset: 0
Keywords
Programs
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Mathematica
With[{m = 24}, Range[0, m]! * CoefficientList[Series[Exp[2*Sin[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *) -
PARI
my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sin(x)))) -
PARI
a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*binomial(n-1, 2*k)*a(n-2*k-1)));
Formula
a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^k * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025