A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 1, 0, 1, 4, 3, 2, 0, 0, 1, 5, 6, 4, 2, 0, 0, 1, 6, 10, 8, 6, 0, 1, 0, 1, 7, 15, 15, 13, 3, 3, 0, 0, 1, 8, 21, 26, 25, 12, 6, 2, 0, 0, 1, 9, 28, 42, 45, 31, 14, 9, 0, 0, 0, 1, 10, 36, 64, 77, 66, 35, 24, 3, 2, 1, 0, 1, 11, 45

Offset: 0

Author

Seiichi Manyama, May 07 2017

Keywords

• nonn
• tabl

Comments

A(n, k) is the number of ways of writing n as the sum of k triangular numbers.

Examples

			Square array begins:
   1, 1, 1, 1,  1,  1, ...
   0, 1, 2, 3,  4,  5, ...
   0, 0, 1, 3,  6, 10, ...
   0, 1, 2, 4,  8, 15, ...
   0, 0, 2, 6, 13, 25, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Columns k=0-12 give A000007, A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787.

Main diagonal gives A106337.

Programs

Mathematica

Table[Function[k, SeriesCoefficient[Product[(1 + x^i) (1 - x^(2 i)), {i, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten (* Michael De Vlieger, May 07 2017 *)

Formula

G.f. of column k: (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k.