%I #5 Mar 07 2021 03:55:55

%S 0,1,2,4,11,16,27,42,69,108,158,229,334,469,656,903,1255,1685,2283,

%T 3032,4033,5290,6936,8986,11650,14969,19172,24402,30998,39110,49260,

%U 61712,77155,96000,119209,147394,181958,223713,274533,335792,409980,498981,606273,734572

%N Total sum of parts which are squares in all partitions of n.

%F G.f.: Sum_{k>=1} k^2*x^(k^2)/(1 - x^(k^2)) / Product_{j>=1} (1 - x^j).

%F a(n) = Sum_{k=1..n} A035316(k) * A000041(n-k).

%e For n = 4 we have:

%e ---------------------------------

%e Partitions Sum of parts

%e . which are squares

%e ---------------------------------

%e 4 ................... 4

%e 3 + 1 ............... 1

%e 2 + 2 ............... 0

%e 2 + 1 + 1 ........... 2

%e 1 + 1 + 1 + 1 ....... 4

%e ---------------------------------

%e Total .............. 11

%e So a(4) = 11.

%t nmax = 43; CoefficientList[Series[Sum[k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]/Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]

%t Table[Sum[DivisorSum[k, # &, IntegerQ[#^(1/2)] &] PartitionsP[n - k], {k, 1, n}], {n, 0, 43}]

%Y Cf. A000041, A000290, A035316, A066186, A073336, A342229.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 06 2021