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%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
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