%I #10 Mar 29 2022 15:10:51

%S 1,1,6,21,82,294,1116,4103,15326,56833,211454,785441,2920058,10851016,

%T 40331874,149892024,557098510,2070493098,7695228038,28600012305,

%U 106294901116,395055313662,1468262641770,5456942875386,20281270503914,75377349437075,280147395367820

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

%C Invert transform of A001157.

%H Seiichi Manyama, <a href="/A320649/b320649.txt">Table of n, a(n) for n = 0..1754</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

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

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

%p 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

%t nmax = 26; CoefficientList[Series[1/(1 - Sum[k^2 x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

%t nmax = 26; CoefficientList[Series[1/(1 + x D[Log[Product[(1 - x^k)^k, {k, 1, nmax}]], x]), {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[2, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 26}]

%Y Cf. A000219, A001157, A180305.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Oct 18 2018