A334191 - OEIS

%I #7 Apr 18 2020 11:49:48

%S 1,0,-3,-9,0,189,1377,4374,-26001,-560601,-4999482,-18631053,

%T 235966365,5966310960,71037580689,407585191059,-3965310883512,

%U -157871090202975,-2631946996862451,-24922384546473810,45577755305571339,7795795206234609027,192159735553383097014

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

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

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

%t 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]

%t nmax = 22; CoefficientList[Series[Exp[x + (1 - Exp[3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Binomial[n, k] * 3^k * BellB[k, -1/3], {k, 0, n}], {n, 0, 22}] (* _Vaclav Kotesovec_, Apr 18 2020 *)

%Y Column k=3 of A334192.

%Y Cf. A003575, A293037, A317996, A334190.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Apr 18 2020