A320652 Expansion of 1/(2 - Product_{k>=1} 1/(1 - k*x^k)).

1, 1, 4, 13, 45, 147, 497, 1643, 5490, 18252, 60812, 202364, 673915, 2243295, 7468973, 24865272, 82783967, 275605513, 917563193, 3054785032, 10170143277, 33858882922, 112724577088, 375287739083, 1249425198725, 4159643200494, 13848474406054, 46104972636634, 153494780854254

Offset: 0

Author

Ilya Gutkovskiy, Oct 18 2018

Keywords

• nonn

Comments

Invert transform of A006906.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A006906, A055887, A257674, A299162.

Programs

Maple

a:=series(1/(2-mul(1/(1-k*x^k),k=1..100)),x=0,29): seq(coeff(a,x,n),n=0..28); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 28; CoefficientList[Series[1/(2 - Product[1/(1 - k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - Sum[k x^k/Product[(1 - j x^j), {j, 1, k}], {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Total[Times@@@IntegerPartitions[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 28}]

Formula

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

a(0) = 1; a(n) = Sum_{k=1..n} A006906(k)*a(n-k).