A320651 Expansion of 1/(1 - Sum_{k>=1} k*x^k/(1 + x^k)).

1, 1, 2, 7, 14, 36, 90, 213, 520, 1271, 3082, 7493, 18238, 44324, 107782, 262142, 637368, 1549870, 3768886, 9164499, 22285034, 54190024, 131771616, 320424614, 779166270, 1894671121, 4607207304, 11203190618, 27242414612, 66244451632, 161084380040, 391703392954

Offset: 0

Author

Ilya Gutkovskiy, Oct 18 2018

Keywords

• nonn

Comments

Invert transform of A000593.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000009, A000593, A180305, A320650.

Programs

Maple

a:=series(1/(1-add(k*x^k/(1+x^k),k=1..100)),x=0,32): seq(coeff(a,x,n),n=0..31); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 31; CoefficientList[Series[1/(1 - Sum[k x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[24/(25 - EllipticTheta[2, 0, x]^4 - EllipticTheta[3, 0, x]^4), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[Mod[d, 2] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 31}]

Formula

G.f.: 1/(1 - x * (d/dx) log(Product_{k>=1} (1 + x^k))).

G.f.: 24/(25 - theta_2(x)^4 - theta_3(x)^4), where theta_() is the Jacobi theta function.

a(0) = 1; a(n) = Sum_{k=1..n} A000593(k)*a(n-k).