A297326 -id:A297326 - cp's OEIS frontend

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.

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

Ilya Gutkovskiy, Dec 28 2017

			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,  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

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.

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

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

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

A297322 a(n) = [x^n] Product_{k>=1} (1 + k*x^k)^n.

Original entry on oeis.org

1, 1, 5, 28, 137, 726, 3896, 21071, 115089, 633007, 3500740, 19448573, 108458924, 606787572, 3404112479, 19142919543, 107874784017, 609021410570, 3443952349385, 19503777943838, 110599636109572, 627924447630011, 3568885868192419, 20304321490356084

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..1000

Main diagonal of A297321.

Cf. A297324, A297326, A297329.

f:= proc(n) local k;
coeff(series(mul((1+k*x^k)^n,k=1..n),x,n+1),x,n);
end proc:
map(f, [$0..30]); # Robert Israel, Dec 28 2017

Table[SeriesCoefficient[Product[(1 + k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]

a(n) = A297321(n,n).

a(n) ~ c * d^n / sqrt(n), where d = 5.814548482877687529318372516965305077397562... and c = 0.2563102401728134539247148322678842806264... - Vaclav Kotesovec, Aug 01 2019

A297329 a(n) = [x^n] Product_{k>=1} 1/(1 - k*x^k)^n.

Original entry on oeis.org

1, 1, 7, 37, 219, 1276, 7687, 46551, 285043, 1756243, 10883842, 67751289, 423366831, 2654041235, 16683909711, 105129718102, 663837626163, 4199521413019, 26610335585263, 168864540960165, 1073001606214814, 6826237566223329, 43474472693256491, 277152041235941803

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Main diagonal of A297328.

Cf. A297322, A297324, A297326.

Table[SeriesCoefficient[Product[1/(1 - k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = A297328(n,n).

a(n) ~ c * d^n / sqrt(n), where d = 6.51548390811914394587815688142024783108478... and c = 0.2552310487728179222346375591994440863074... - Vaclav Kotesovec, Aug 01 2019

A297324 a(n) = [x^n] Product_{k>=1} (1 - k*x^k)^n.

Original entry on oeis.org

1, -1, -3, 8, 9, -26, -168, 489, 1041, -5599, 12, 27103, 23436, -222912, -435473, 3177433, 375569, -24956018, 6931209, 181844002, 57372644, -2158209675, 853739235, 20642183588, -25063980804, -148768035501, 224915906836, 1322267927471, -2337343745721, -12604818831294

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Main diagonal of A297323.

Cf. A297322, A297326, A297329.

Table[SeriesCoefficient[Product[(1 - k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 29}]

a(n) = A297323(n,n).

A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.

Original entry on oeis.org

1, -1, 0, -18, 208, -2400, 36504, -663754, 13808320, -324176418, 8487126400, -245122390601, 7741417124880, -265402847130421, 9816338228638872, -389618889514254225, 16518399076342421248, -745025763154442071130, 35619835529954597786208, -1799459812004380374518790, 95780758238408017088795600

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

sign

Cf. A255528, A261566, A261567, A266971, A292134, A297326, A298985, A298986, A298987.

Table[SeriesCoefficient[Product[1/(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

a(n) ~ (-1)^n * n^n * (1 - 2/n + 6/n^2 - 14/n^3 + 33/n^4 - 70/n^5 + 149/n^6 - 298/n^7 + 591/n^8 - 1132/n^9 + 2139/n^10 + ...), for coefficients, see A005380. - Vaclav Kotesovec, Aug 21 2018

cp's OEIS Frontend

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.

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A297322 a(n) = [x^n] Product_{k>=1} (1 + k*x^k)^n.

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A297329 a(n) = [x^n] Product_{k>=1} 1/(1 - k*x^k)^n.

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A297324 a(n) = [x^n] Product_{k>=1} (1 - k*x^k)^n.

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A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.

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