A357727 Expansion of e.g.f. cos( 2 * (exp(x) - 1) ).

1, 0, -4, -12, -12, 100, 852, 4004, 9940, -36828, -726316, -6174300, -35968812, -109708508, 702818004, 16677814436, 188794428628, 1542659688996, 8359981681364, -3068614764636, -868989327994668, -15076627082974940, -179727483880747308

Offset: 0

Author

Seiichi Manyama, Oct 10 2022

Keywords

• sign

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

Column k=4 of A357728.

Cf. A065143, A357719, A357738.

Programs

PARI

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

PARI

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

PARI

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 2*I)+Bell_poly(n, -2*I)))/2;

Formula

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

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

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.