A307258 Expansion of (1/(1 + x)) * Product_{k>=1} 1/(1 - k*x^k/(1 + x)^k).

1, 0, 2, -1, 5, -11, 36, -107, 311, -850, 2208, -5519, 13566, -33562, 84937, -220307, 579413, -1522616, 3954016, -10100863, 25416877, -63324271, 157248035, -391478354, 980410093, -2470810086, 6253495883, -15846525758, 40093721908, -101116823798, 254093749587, -636547773777

Offset: 0

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

• sign

Comments

Inverse binomial transform of A006906.

Crossrefs

Cf. A006906, A281425, A294501, A307260, A318127.

Programs

Maple

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

Mathematica

nmax = 31; CoefficientList[Series[1/(1 + x) Product[1/(1 - k x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[n, k] Total[Times @@@ IntegerPartitions[k]], {k, 0, n}], {n, 0, 31}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A006906(k).