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).

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

Offset: 0

Author

Emeric Deutsch, Dec 06 2015

Keywords

• nonn
• tabf

Comments

Number of entries in row n = 1 + n^2.

Sum of entries in row n = A000041(n).

Sum(k*T(n,k), k>=0) = A066183(n).

Examples

			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.
		

Links

• Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Crossrefs

Cf. A000041, A066183, A229325 - A229332, A264402, A265247 - A265253.

Programs

Maple

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

Mathematica

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 *)

Formula

G.f.: G(t,x) = 1/Product_{k>=1} (1 - t^{k^2}*x^k).