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A357667 Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).

%I A357667 #29 Feb 16 2025 08:34:04
%S A357667 1,0,9,27,144,945,6273,44226,339399,2796795,24387786,223853355,
%T A357667 2159078445,21827316888,230536050165,2536213188519,28994911890048,
%U A357667 343806474384045,4220933769308205,53566838971016418,701650841036287275,9473067208871584407
%N A357667 Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).
%H A357667 Seiichi Manyama, <a href="/A357667/b357667.txt">Table of n, a(n) for n = 0..547</a>
%H A357667 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357667 E.g.f.: cosh( 3 * (exp(x) - 1) ).
%F A357667 a(n) = Sum_{k=0..floor(n/2)} 9^k * Stirling2(n,2*k).
%F A357667 a(n) = ( Bell_n(3) + Bell_n(-3) )/2, where Bell_n(x) is n-th Bell polynomial.
%F A357667 a(n) = 1; a(n) = 9 * Sum_{k=0..n-1} binomial(n-1, k) * A357668(k).
%o A357667 (PARI) my(x='x+O('x^30)); Vec(serlaplace(cosh(3*(exp(x)-1))))
%o A357667 (PARI) a(n) = sum(k=0, n\2, 9^k*stirling(n, 2*k, 2));
%o A357667 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357667 a(n) = round((Bell_poly(n, 3)+Bell_poly(n, -3)))/2;
%Y A357667 Column k=9 of A357681.
%Y A357667 Cf. A024430, A065143, A264036, A357615.
%Y A357667 Cf. A027710, A357649, A357668.
%K A357667 nonn
%O A357667 0,3
%A A357667 _Seiichi Manyama_, Oct 08 2022