A357726 - OEIS

%I #23 Jan 31 2024 04:05:35

%S 1,0,-3,-9,-12,45,465,2394,7827,639,-250410,-2588553,-17773635,

%T -84525480,-105849399,3569654115,56100280308,561682625769,

%U 4227837863181,20472943653306,-38990802816489,-2621206974761253,-42512769453705474,-495174030273565173

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

%H Andrew Howroyd, <a href="/A357726/b357726.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)} (-3)^k * Stirling2(n,2*k).

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

%F 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.

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

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

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

%Y Column k=3 of A357728.

%Y Cf. A357615, A357737.

%K sign

%O 0,3

%A _Seiichi Manyama_, Oct 10 2022