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A297325
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + j*x^j)^k.
15
1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, -2, 0, 1, -4, 0, -2, 2, 0, 1, -5, 2, -1, 9, -1, 0, 1, -6, 5, 0, 18, -2, 4, 0, 1, -7, 9, 0, 27, -12, 10, -1, 0, 1, -8, 14, -2, 35, -36, 11, -16, 18, 0, 1, -9, 20, -7, 42, -76, 14, -54, 38, -22, 0, 1, -10, 27, -16, 49, -132, 35, -104, 84, -98, 12, 0
OFFSET
0,8
LINKS
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
KEYWORD
sign,tabl
AUTHOR
Ilya Gutkovskiy, Dec 28 2017
STATUS
approved