A297321 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j*x^j)^k.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 7, 0, 1, 5, 14, 28, 28, 15, 0, 1, 6, 20, 48, 69, 64, 25, 0, 1, 7, 27, 75, 137, 174, 133, 43, 0, 1, 8, 35, 110, 240, 380, 413, 266, 64, 0, 1, 9, 44, 154, 387, 726, 998, 933, 513, 120, 0, 1, 10, 54, 208, 588, 1266, 2075, 2488, 2046, 1000, 186, 0

Offset

0,8

Links

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened

Formula

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

Example

G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 3)*x^2 + (1/6)*k*(k^2 + 9*k + 20)*x^3 + (1/24)*k*(k^3 + 18*k^2 + 107*k + 42)*x^4 + (1/120)*k*(k^4 + 30*k^3 + 335*k^2 + 810*k + 624)*x^5 + ...
Square array begins:
1,   1,   1,    1,    1,    1,  ...
0,   1,   2,    3,    4,    5,  ...
0,   2,   5,    9,   14,   20,  ...
0,   5,  14,   28,   48,   75,  ...
0,   7,  28,   69,  137,  240,  ...
0,  15,  64,  174,  380,  726,  ...

Mathematica

Table[Function[k, SeriesCoefficient[Product[(1 + i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0..32 give A000007, A022629, A022630, A022631, A022632, A022633, A022634, A022635, A022636, A022637, A022638, A022639, A022640, A022641, A022642, A022643, A022644, A022645, A022646, A022647, A022648, A022649, A022650, A022651, A022652, A022653, A022654, A022655, A022656, A022657, A022658, A022659, A022660.

Main diagonal gives A297322.

Antidiagonal sums give A299164.

Cf. A266891, A297323, A297325, A297328.

Sequence in context: A144064 A172236 A191646 * A277938 A130020 A292870

Adjacent sequences: A297318 A297319 A297320 * A297322 A297323 A297324

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Dec 28 2017

Status

approved