OFFSET
0,8
LINKS
Alois P. Heinz, Antidiagonals n = 0..200, flattened
FORMULA
G.f. of column k: Product_{j>=1} 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, -1, -1, 0, 2, 5, ...
0, -2, -2, -1, 0, 0, ...
0, 2, 9, 18, 27, 35, ...
0, -1, -2, -12, -36, -76, ...
MAPLE
with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, -k*add(add(
(-d)^(1+j/d), d=divisors(j))*A(n-j, k), j=1..n)/n)
end:
seq(seq(A(n, d-n), n=0..d), d=0..14); # Alois P. Heinz, Apr 20 2018
MATHEMATICA
Table[Function[k, SeriesCoefficient[Product[1/(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, A022693, A022694, A022695, A022696, A022697, A022698, A022699, A022700, A022701, A022702, A022703, A022704, A022705, A022706, A022707, A022708, A022709, A022710, A022711, A022712, A022713, A022714, A022715, A022716, A022717, A022718, A022719, A022720, A022721, A022722, A022723, A022724.
Main diagonal gives A297326.
Antidiagonal sums give A299210.
KEYWORD
sign,tabl
AUTHOR
Ilya Gutkovskiy, Dec 28 2017
STATUS
approved