A290430 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j*(j+1)*(2*j+1)/6))^k.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 1, 0, 1, 6, 10, 4, 0, 2, 0, 0, 1, 7, 15, 10, 1, 3, 2, 0, 0, 1, 8, 21, 20, 5, 4, 6, 0, 0, 0, 1, 9, 28, 35, 15, 6, 12, 3, 0, 0, 0, 1, 10, 36, 56, 35, 12, 20, 12, 0, 0, 0, 0, 1, 11, 45, 84, 70, 28, 31, 30, 4, 0, 1, 0, 0, 1, 12, 55, 120, 126, 64, 49, 60, 20, 0, 3, 0, 0, 0

Offset: 0

Author

Ilya Gutkovskiy, Jul 31 2017

Keywords

• nonn
• tabl

Comments

A(n,k) is the number of ways of writing n as a sum of k square pyramidal numbers (A000330).

Examples

			Square array begins:
1,  1,  1,  1,  1,   1,  ...
0,  1,  2,  3,  4,   5,  ...
0,  0,  1,  3,  6,  10,  ...
0,  0,  0,  1,  4,  10,  ...
0,  0,  0,  0,  1,   5,  ...
0,  1,  2,  3,  4,   6,  ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened
• Eric Weisstein's World of Mathematics, Square Pyramidal Number
• Index to sequences related to pyramidal numbers

Crossrefs

Cf. A000330, A045847, A122141, A286815, A290429.

Cf. A000007 (column 0), A253903 (column 1), A282173 (column 6).

Main diagonal gives A303172.

Programs

Mathematica

Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (2 i + 1)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

Formula

G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(2*j+1)/6))^k.