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A265245 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the squares of the parts is k (n>=0, k>=0).

%I A265245 #14 Feb 19 2019 09:21:18
%S A265245 1,0,1,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,
%T A265245 1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,
%U A265245 0,1,0,1,0,2,0,1,0,0,0,2,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,1,0,2,0,1,0,1,0,2,0,1,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1
%N A265245 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the squares of the parts is k (n>=0, k>=0).
%C A265245 Number of entries in row n = 1 + n^2.
%C A265245 Sum of entries in row n = A000041(n).
%C A265245 Sum(k*T(n,k), k>=0) = A066183(n).
%H A265245 Guo-Niu Han, <a href="http://arxiv.org/abs/0804.1849">An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths</a>, arXiv:0804.1849 [math.CO], 2008.
%F A265245 G.f.: G(t,x) = 1/Product_{k>=1} (1 - t^{k^2}*x^k).
%e A265245 Row 3 is 0,0,0,1,0,1,0,0,0,1 because in the partitions of 3, namely [1,1,1], [2,1], [3], the sums of the squares of the parts are 3, 5, and 9, respectively.
%e A265245 Triangle starts:
%e A265245 1;
%e A265245 0,1;
%e A265245 0,0,1,0,1;
%e A265245 0,0,0,1,0,1,0,0,0,1;
%e A265245 0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1.
%p A265245 g := 1/(product(1-t^(k^2)*x^k, k = 1 .. 100)): gser := simplify(series(g, x = 0, 15)): for n from 0 to 8 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 8 do seq(coeff(P[n], t, j), j = 0 .. n^2) end do; # yields sequence in triangular form
%t A265245 m = 8; CoefficientList[#, t]& /@ CoefficientList[1/Product[(1 - t^(k^2)* x^k), {k, 1, m}] + O[x]^m, x] // Flatten (* _Jean-François Alcover_, Feb 19 2019 *)
%Y A265245 Cf. A000041, A066183, A229325 - A229332, A264402, A265247 - A265253.
%K A265245 nonn,tabf
%O A265245 0,74
%A A265245 _Emeric Deutsch_, Dec 06 2015