A357726 Expansion of e.g.f. cos( sqrt(3) * (exp(x) - 1) ).

1, 0, -3, -9, -12, 45, 465, 2394, 7827, 639, -250410, -2588553, -17773635, -84525480, -105849399, 3569654115, 56100280308, 561682625769, 4227837863181, 20472943653306, -38990802816489, -2621206974761253, -42512769453705474, -495174030273565173

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=3 of A357728.

Cf. A357615, A357737.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Cos[Sqrt[3](Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 20 2023 *)

PARI

my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(3)*(exp(x)-1)))))

PARI

a(n) = sum(k=0, n\2, (-3)^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, sqrt(3)*I)+Bell_poly(n, -sqrt(3)*I)))/2;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-3)^k * Stirling2(n,2*k).

a(n) = 1; a(n) = -3 * Sum_{k=0..n-1} binomial(n-1, k) * A357737(k).

a(n) = ( Bell_n(sqrt(3) * i) + Bell_n(-sqrt(3) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.