A299209 -id:A299209 - cp's OEIS frontend

A297323 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^k.

1, 1, 0, 1, -1, 0, 1, -2, -2, 0, 1, -3, -3, -1, 0, 1, -4, -3, 2, -1, 0, 1, -5, -2, 8, 4, 5, 0, 1, -6, 0, 16, 9, 16, 1, 0, 1, -7, 3, 25, 9, 18, -3, 13, 0, 1, -8, 7, 34, 0, 4, -35, 6, 4, 0, 1, -9, 12, 42, -21, -26, -90, -33, -31, 0, 0, 1, -10, 18, 48, -56, -66, -145, -56, -66, -72, 2, 0

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

			G.f. of column k: A_k(x) = 1 - k*x + (1/2)*k*(k - 5)*x^2 - (1/6)*k*(k^2 - 15*k + 20)*x^3 + (1/24)*k*(k^3 - 30*k^2 + 155*k - 150)*x^4 - (1/120)*k*(k^4 - 50*k^3 + 575*k^2 - 1750*k + 624)*x^5 + ...
Square array begins:
  1,  1,   1,   1,   1,   1,  ...
  0, -1,  -2,  -3,  -4,  -5,  ...
  0, -2,  -3,  -3,  -2,   0,  ...
  0, -1,   2,   8,  16,  25,  ...
  0, -1,   4,   9,   9,   0,  ...
  0,  5,  16,  18,   4, -26,  ...

Columns k=0..32 give A000007, A022661, A022662, A022663, A022664, A022665, A022666, A022667, A022668, A022669, A022670, A022671, A022672, A022673, A022674, A022675, A022676, A022677, A022678, A022679, A022680, A022681, A022682, A022683, A022684, A022685, A022686, A022687, A022688, A022689, A022690, A022691, A022692.
Main diagonal gives A297324.
Antidiagonal sums give A299209.
Cf. A266964, A297321, A297325, A297328.

Table[Function[k, SeriesCoefficient[Product[(1 - i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, (1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ Iain Fox, Dec 28 2017

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

A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, 1, 0, -1, -2, -1, 1, 3, 3, 1, -3, -6, -5, 1, 9, 12, 5, -9, -20, -18, 1, 26, 38, 21, -21, -61, -62, -9, 72, 120, 81, -44, -177, -205, -64, 186, 366, 293, -63, -496, -657, -304, 445, 1084, 1014, 33, -1341, -2053, -1238, 959, 3132, 3378, 770, -3474, -6260, -4619, 1656, 8809, 10929, 4306, -8520

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286352.
Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A081362, A299108, A299162, A299164, A299166, A299167, A299209, A299210, A299211, A299212.

nmax = 60; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(0) = 1; a(n) = Sum_{k=1..n} A081362(k-1)*a(n-k).

A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

Original entry on oeis.org

1, 1, 0, -3, -6, -4, 12, 39, 52, -9, -186, -392, -285, 610, 2291, 3200, -150, -10626, -23487, -18841, 32957, 134848, 198246, 13961, -605248, -1409604, -1234474, 1744213, 7898753, 12209679, 2161666, -34344627, -84393284, -79993042, 90692470, 461463974, 749309529, 207447895, -1939084232

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

Robert Israel, Table of n, a(n) for n = 0..3925
N. J. A. Sloane, Transforms

Antidiagonal sums of A276554.

Cf. A067687, A073592, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299212.

N:= 100: # for a(0)..a(N)
S:= series(1/(1-x*mul((1-x^k)^k,k=1..N)),x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Feb 05 2023

nmax = 38; CoefficientList[Series[1/(1 - x Product[(1 - x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

a(0) = 1; a(n) = Sum_{k=1..n} A073592(k-1)*a(n-k).

A299210 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 + kx^k)).

Original entry on oeis.org

1, 1, 0, -2, -5, -3, 5, 20, 27, 17, -53, -152, -192, 31, 576, 1110, 694, -1297, -4519, -6160, -1107, 13665, 31914, 30643, -19339, -119260, -196142, -103318, 289543, 859631, 1062684, 13710, -2690348, -5675946, -4940757, 4167527, 21343918, 33874107, 16524162, -51704908, -150454546

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

N. J. A. Sloane, Transforms

Antidiagonal sums of A297325.

Cf. A022693, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299211, A299212.

nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(0) = 1; a(n) = Sum_{k=1..n} A022693(k-1)*a(n-k).

A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

Original entry on oeis.org

1, 1, 0, -2, -5, -4, 4, 21, 35, 23, -47, -165, -239, -78, 479, 1273, 1508, -138, -4429, -9451, -8845, 6207, 37937, 67123, 45144, -83355, -308078, -455109, -166872, 873799, 2393041, 2916869, -73472, -8133572, -17828640, -17294146, 10383571, 70275162, 127401305, 90368779, -147825714

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

N. J. A. Sloane, Transforms

Antidiagonal sums of A279928.

Cf. A067687, A255528, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299211.

nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(0) = 1; a(n) = Sum_{k=1..n} A255528(k-1)*a(n-k).

cp's OEIS Frontend

A297323 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^k.

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A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

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A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

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A299210 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).

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A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

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A299210 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 + kx^k)).