A318127 Expansion of (1/(1 - x)) * Product_{k>=1} 1/(1 - k*x^k/(1 - x)^k).

1, 2, 6, 19, 61, 191, 588, 1785, 5351, 15868, 46628, 135921, 393318, 1130538, 3229753, 9175347, 25931605, 72936434, 204223348, 569427145, 1581458917, 4375905243, 12065914843, 33160240020, 90848002909, 248154744196, 675932128695, 1836182233332, 4975249827916, 13447775233746

Offset: 0

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

• nonn

Comments

Binomial transform of A006906.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A006906, A218481, A294500.

Programs

Maple

a:=series(1/(1-x)*mul(1/(1-k*x^k/(1-x)^k),k=1..100),x=0,30): seq(coeff(a,x,n),n=0..29); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 29; CoefficientList[Series[1/(1 - x) Product[1/(1 - k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[j^k x^(k j)/(k (1 - x)^(k j)), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[Binomial[n, k] Total[Times @@@ IntegerPartitions[k]], {k, 0, n}], {n, 0, 29}]

Formula

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

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

a(n) ~ c * (1 + 3^(1/3))^n, where c = 97923.037496367052161042295948902147352859984491653037730624387144966464... = 1/((3^(1/3) - 1) * (3^(2/3) - 2)) * Product_{k>=4} 1/(1 - k/3^(k/3)). - Vaclav Kotesovec, Aug 19 2018