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A297325
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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.
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15
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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
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OFFSET
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0,8
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..200, flattened
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FORMULA
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G.f. of column k: Product_{j>=1} 1/(1 + j*x^j)^k.
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EXAMPLE
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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, ...
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MAPLE
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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
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MATHEMATICA
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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
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CROSSREFS
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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.
Cf. A266971, A297321, A297323, A297328.
Sequence in context: A047265 A185962 A279928 * A278528 A257261 A355141
Adjacent sequences: A297322 A297323 A297324 * A297326 A297327 A297328
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KEYWORD
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sign,tabl
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AUTHOR
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Ilya Gutkovskiy, Dec 28 2017
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STATUS
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approved
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