A320649 Expansion of 1/(1 - Sum_{k>=1} k^2*x^k/(1 - x^k)).

1, 1, 6, 21, 82, 294, 1116, 4103, 15326, 56833, 211454, 785441, 2920058, 10851016, 40331874, 149892024, 557098510, 2070493098, 7695228038, 28600012305, 106294901116, 395055313662, 1468262641770, 5456942875386, 20281270503914, 75377349437075, 280147395367820

Offset: 0

Author

Ilya Gutkovskiy, Oct 18 2018

Keywords

• nonn

Comments

Invert transform of A001157.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1754
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000219, A001157, A180305.

Programs

Maple

a:=series(1/(1-add(k^2*x^k/(1-x^k),k=1..100)),x=0,27): seq(coeff(a,x,n),n=0..26); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 26; CoefficientList[Series[1/(1 - Sum[k^2 x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 26; CoefficientList[Series[1/(1 + x D[Log[Product[(1 - x^k)^k, {k, 1, nmax}]], x]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[2, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 26}]

Formula

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

a(0) = 1; a(n) = Sum_{k=1..n} sigma_2(k)*a(n-k).