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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 (list; table; graph; refs; listen; history; text; internal format)
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.

Cf. A266971, A297321, A297323, A297328.

Sequence in context: A047265 A185962 A279928 * A278528 A257261 A355141

Adjacent sequences:  A297322 A297323 A297324 * A297326 A297327 A297328

KEYWORD

sign,tabl

AUTHOR

Ilya Gutkovskiy, Dec 28 2017

STATUS

approved

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Last modified August 15 02:47 EDT 2022. Contains 356122 sequences. (Running on oeis4.)