login
Expansion of e.g.f. sinh( 3 * (exp(x) - 1) )/3.
2

%I #24 Jan 25 2024 12:47:31

%S 0,1,1,10,55,307,2026,14779,114157,933616,8110261,74525167,719925328,

%T 7279859485,76855303981,845280487018,9663800287483,114601481983855,

%U 1407040763488354,17856103120048783,233883061849700137,3157648445216335528,43887908697233605489

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

%H Seiichi Manyama, <a href="/A357668/b357668.txt">Table of n, a(n) for n = 0..547</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-1)/2)} 9^k * Stirling2(n,2*k+1).

%F a(n) = ( Bell_n(3) - Bell_n(-3) )/6, where Bell_n(x) is n-th Bell polynomial.

%F a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357667(k).

%o (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(3*(exp(x)-1))/3)))

%o (PARI) a(n) = sum(k=0, (n-1)\2, 9^k*stirling(n, 2*k+1, 2));

%o (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);

%o a(n) = round((Bell_poly(n, 3)-Bell_poly(n, -3)))/6;

%Y Cf. A024429, A264037, A357572, A357598.

%Y Cf. A027710, A356572, A357667.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 08 2022