A334191 a(n) = exp(1/3) * Sum_{k>=0} (3*k + 1)^n / ((-3)^k * k!).

1, 0, -3, -9, 0, 189, 1377, 4374, -26001, -560601, -4999482, -18631053, 235966365, 5966310960, 71037580689, 407585191059, -3965310883512, -157871090202975, -2631946996862451, -24922384546473810, 45577755305571339, 7795795206234609027, 192159735553383097014

Offset: 0

Author

Ilya Gutkovskiy, Apr 18 2020

Keywords

• sign

Crossrefs

Column k=3 of A334192.

Cf. A003575, A293037, A317996, A334190.

Programs

Mathematica

nmax = 22; CoefficientList[Series[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - 3 j x/(1 - x)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Exp[x + (1 - Exp[3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] * 3^k * BellB[k, -1/3], {k, 0, n}], {n, 0, 22}] (* Vaclav Kotesovec, Apr 18 2020 *)

Formula

G.f.: (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - 3*j*x/(1 - x)).

E.g.f.: exp(x + (1 - exp(3*x)) / 3).