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

0,8

Comments

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

Links

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

Eric Weisstein's World of Mathematics, Square Pyramidal Number

Index to sequences related to pyramidal numbers

Formula

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

Example

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,  ...

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

Crossrefs

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

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

Main diagonal gives A303172.

Sequence in context: A077029 A052553 A290054 * A290429 A045847 A137586

Adjacent sequences: A290427 A290428 A290429 * A290431 A290432 A290433

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Jul 31 2017

Status

approved