A357667 Expansion of e.g.f. cosh( 3 * (exp(x) - 1) ).
1, 0, 9, 27, 144, 945, 6273, 44226, 339399, 2796795, 24387786, 223853355, 2159078445, 21827316888, 230536050165, 2536213188519, 28994911890048, 343806474384045, 4220933769308205, 53566838971016418, 701650841036287275, 9473067208871584407
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..547
- Eric Weisstein's World of Mathematics, Bell Polynomial.
Crossrefs
Programs
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PARI
my(x='x+O('x^30)); Vec(serlaplace(cosh(3*(exp(x)-1)))) -
PARI
a(n) = sum(k=0, n\2, 9^k*stirling(n, 2*k, 2));
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PARI
Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!); a(n) = round((Bell_poly(n, 3)+Bell_poly(n, -3)))/2;
Formula
E.g.f.: cosh( 3 * (exp(x) - 1) ).
a(n) = Sum_{k=0..floor(n/2)} 9^k * Stirling2(n,2*k).
a(n) = ( Bell_n(3) + Bell_n(-3) )/2, where Bell_n(x) is n-th Bell polynomial.
a(n) = 1; a(n) = 9 * Sum_{k=0..n-1} binomial(n-1, k) * A357668(k).