%I #8 Apr 03 2019 09:03:55
%S 1,0,2,-1,5,-11,36,-107,311,-850,2208,-5519,13566,-33562,84937,
%T -220307,579413,-1522616,3954016,-10100863,25416877,-63324271,
%U 157248035,-391478354,980410093,-2470810086,6253495883,-15846525758,40093721908,-101116823798,254093749587,-636547773777
%N Expansion of (1/(1 + x)) * Product_{k>=1} 1/(1 - k*x^k/(1 + x)^k).
%C Inverse binomial transform of A006906.
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A006906(k).
%p a:=series((1/(1+x))*mul(1/(1-k*x^k/(1+x)^k),k=1..100),x=0,32): seq(coeff(a,x,n),n=0..31); # _Paolo P. Lava_, Apr 03 2019
%t nmax = 31; CoefficientList[Series[1/(1 + x) Product[1/(1 - k x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t Table[Sum[(-1)^(n - k) Binomial[n, k] Total[Times @@@ IntegerPartitions[k]], {k, 0, n}], {n, 0, 31}]
%Y Cf. A006906, A281425, A294501, A307260, A318127.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Apr 01 2019