A320354 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Product_{j=1..n} (k^j - 1).

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 16, 21, 0, 1, 4, 45, 416, 315, 0, 1, 5, 96, 2835, 33280, 9765, 0, 1, 6, 175, 11904, 722925, 8053760, 615195, 0, 1, 7, 288, 37625, 7428096, 739552275, 5863137280, 78129765, 0, 1, 8, 441, 98496, 48724375, 23205371904, 3028466566125, 12816818094080, 19923090075, 0

Offset: 0

Author

Ilya Gutkovskiy, Oct 11 2018

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,     1,        1,          1,            1,             1,  ...
  0,     1,        2,          3,            4,             5,  ...
  0,     3,       16,         45,           96,           175,  ...
  0,    21,      416,       2835,        11904,         37625,  ...
  0,   315,    33280,     722925,      7428096,      48724375,  ...
  0,  9765,  8053760,  739552275,  23205371904,  378832015625,  ...
		

Crossrefs

Columns k=1..12 give A000007, A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880.

Cf. A069777, A225816.

Programs

Mathematica

Table[Function[k, Product[k^j - 1, {j, 1, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[k^(i (i + 1)/2) x^i/Product[(1 + k^j x), {j, 0, i}], {i, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten

Formula

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

For asymptotics of column k see comment from Vaclav Kotesovec in A027880.