A335813 - OEIS

%I #11 Jul 09 2020 02:46:56

%S 1,-1,1,-7,-11,-151,-419,-1807,-5291,-381031,-9125939,-139879807,

%T -1217973371,7055720489,657464911741,20268419534993,455079458957749,

%U 7487596915540409,62151133224856621,-943454812059725407,-32387452121872219931,1120264679544729734729

%N Expansion of e.g.f. Product_{k>=1} (1 + (1 - exp(x))^k).

%C Inverse binomial transform of A335811.

%F a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A000009(k).

%t nmax = 21; CoefficientList[Series[Product[(1 + (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[(-1)^k StirlingS2[n, k] k! PartitionsQ[k], {k, 0, n}], {n, 0, 21}]

%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+(1-exp(x))^k))) \\ _Seiichi Manyama_, Jul 08 2020

%Y Cf. A000009, A305550, A335811, A335812.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Jun 25 2020