A367744 Expansion of e.g.f. exp(1 - x - exp(3*x)).
1, -4, 7, 17, -14, -637, -2951, 14126, 333205, 2076245, -12283700, -423234511, -4163106203, 8148184700, 952894223755, 15568620884189, 69314620864450, -2816256959131561, -83397946135434515, -1025683419252783946, 4726361848234575553, 525779836596438636689, 12363747028673287330948, 112888493670408785796989
Offset: 0
Keywords
Programs
-
Mathematica
nmax = 23; CoefficientList[Series[Exp[1 - x - Exp[3 x]], {x, 0, nmax}], x] Range[0, nmax]! a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}] Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k BellB[k, -1], {k, 0, n}], {n, 0, 23}]
Formula
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (3*k-1)^n / k!.
a(0) = 1; a(n) = -a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * A000587(k).