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Expansion of e.g.f. sin( 2 * (exp(x) - 1) )/2.
1

%I #17 Feb 19 2023 09:55:05

%S 0,1,1,-3,-23,-83,-119,973,11145,69805,278281,33165,-12794231,

%T -157150355,-1271714807,-7108146611,-11364216951,380051588653,

%U 6923479542025,78935931180813,669998027706505,3602978599128301,-8825050911646199,-598024924863875123

%N Expansion of e.g.f. sin( 2 * (exp(x) - 1) )/2.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%F a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^(k) * Stirling2(n,2*k+1).

%F a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357727(k).

%F a(n) = ( Bell_n(2 * i) - Bell_n(-2 * i) )/(4 * i), where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

%t With[{nn=30},CoefficientList[Series[Sin[2(Exp[x]-1)]/2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 19 2023 *)

%o (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sin(2*(exp(x)-1))/2)))

%o (PARI) a(n) = sum(k=0, (n-1)\2, (-4)^k*stirling(n, 2*k+1, 2));

%o (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);

%o a(n) = round((Bell_poly(n, 2*I)-Bell_poly(n, -2*I)))/(4*I);

%Y Cf. A357598, A357727.

%K sign

%O 0,4

%A _Seiichi Manyama_, Oct 11 2022