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A265245
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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).
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1
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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, 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, 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
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OFFSET
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0,74
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COMMENTS
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Number of entries in row n = 1 + n^2.
Sum of entries in row n = A000041(n).
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LINKS
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FORMULA
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G.f.: G(t,x) = 1/Product_{k>=1} (1 - t^{k^2}*x^k).
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EXAMPLE
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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.
Triangle starts:
1;
0,1;
0,0,1,0,1;
0,0,0,1,0,1,0,0,0,1;
0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1.
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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