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A357615 Expansion of e.g.f. cosh(sqrt(3) * (exp(x) - 1)).

%I A357615 #25 Feb 16 2025 08:34:04
%S A357615 1,0,3,9,30,135,705,3906,22953,145053,985800,7136613,54544485,
%T A357615 437961888,3685605735,32441696325,297977767662,2848636972971,
%U A357615 28278241848309,290931124989546,3097051613077269,34064462020306473,386600759467746528,4521440483724439521
%N A357615 Expansion of e.g.f. cosh(sqrt(3) * (exp(x) - 1)).
%H A357615 Seiichi Manyama, <a href="/A357615/b357615.txt">Table of n, a(n) for n = 0..562</a>
%H A357615 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357615 a(n) = Sum_{k=0..floor(n/2)} 3^k * Stirling2(n,2*k).
%F A357615 a(n) = ( Bell_n(sqrt(3)) + Bell_n(-sqrt(3)) )/2, where Bell_n(x) is n-th Bell polynomial.
%F A357615 a(n) = 1; a(n) = 3 * Sum_{k=0..n-1} binomial(n-1, k) * A357572(k).
%e A357615 G.f. = 1 + 3*x^2 + 9*x^3 + 30*x^4 + 135*x^5 + 705*x^6 + ... - _Michael Somos_, Oct 06 2022
%t A357615 a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Cosh[Sqrt[3] * (Exp@x - 1)], {x, 0, n}]]; (* _Michael Somos_, Oct 06 2022 *)
%o A357615 (PARI) a(n) = sum(k=0, n\2, 3^k*stirling(n, 2*k, 2));
%o A357615 (PARI) my(x='x+O('x^30)); apply(round, Vec(serlaplace(cosh(sqrt(3) * (exp(x) - 1))))) \\ _Michel Marcus_, Oct 06 2022
%o A357615 (PARI)  {a(n) = if(n<0, 0, n!*simplify(polcoeff( cosh(quadgen(12) * (exp(x + x*O(x^n)) - 1)), n)))}; /* _Michael Somos_, Oct 06 2022 */
%Y A357615 Column k=3 of A357681.
%Y A357615 Cf. A065143, A264036.
%Y A357615 Cf. A357572.
%K A357615 nonn
%O A357615 0,3
%A A357615 _Seiichi Manyama_, Oct 06 2022