%I #22 Jan 31 2024 04:05:42

%S 1,0,-4,-12,-12,100,852,4004,9940,-36828,-726316,-6174300,-35968812,

%T -109708508,702818004,16677814436,188794428628,1542659688996,

%U 8359981681364,-3068614764636,-868989327994668,-15076627082974940,-179727483880747308

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

%H Andrew Howroyd, <a href="/A357727/b357727.txt">Table of n, a(n) for n = 0..200</a>

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

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

%F a(n) = 1; a(n) = -4 * Sum_{k=0..n-1} binomial(n-1, k) * A357738(k).

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

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

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

%Y Column k=4 of A357728.

%Y Cf. A065143, A357719, A357738.

%K sign

%O 0,3

%A _Seiichi Manyama_, Oct 10 2022