A299108 -id:A299108 - cp's OEIS frontend

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

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).

A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 8, 0, 1, 8, 24, 32, 14, 0, 1, 10, 40, 80, 76, 24, 0, 1, 12, 60, 160, 234, 168, 40, 0, 1, 14, 84, 280, 552, 624, 352, 64, 0, 1, 16, 112, 448, 1110, 1712, 1552, 704, 100, 0, 1, 18, 144, 672, 2004, 3912, 4896, 3648, 1356, 154, 0, 1, 20, 180, 960, 3346, 7896, 12600, 13120, 8184, 2532, 232, 0

Offset: 0

Ilya Gutkovskiy, Jun 10 2017

			Square array begins:
1,   1,    1,    1,     1,     1,  ...
0,   2,    4,    6,     8,    10,  ...
0,   4,   12,   24,    40,    60,  ...
0,   8,   32,   80,   160,   280,  ...
0,  14,   76,  234,   552,  1110,  ...
0,  24,  168,  624,  1712,  3913,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=0-24 give: A000007, A015128, A001934, A004404 (alternating values), A284286, A004406-A004425 (alternating values).
Rows n=0-2 give: A000012, A005843, A046092.
Main diagonal gives A270919.
Antidiagonal sums give A299108.
Cf. A122141, A286815.

# JacobiTheta4 is defined in A002448.
A288515Column(k, len) = JacobiTheta4(len, -k)
for k in 0:8 A288515Column(k, 8) |> println end # Peter Luschny, Mar 12 2018

Table[Function[k, SeriesCoefficient[Product[((1 + x^i)/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/EllipticTheta[4, 0, x]^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
G.f. of column k: 1/theta_4(x)^k, where theta_4() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A001934.

A302020 Expansion of 1/(1 - xProduct_{k>=1} (1 + x^(2k))/(1 - x^(2*k-1))).

Original entry on oeis.org

1, 1, 2, 5, 12, 28, 66, 156, 367, 863, 2031, 4779, 11244, 26456, 62248, 146462, 344608, 810822, 1907769, 4488757, 10561519, 24850017, 58469179, 137571128, 323688747, 761601701, 1791959579, 4216270956, 9920391613, 23341519267, 54919860316, 129219997322, 304039515247, 715369360371

Offset: 0

Ilya Gutkovskiy, Mar 30 2018

nonn

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Partition Function b_k

Antidiagonal sums of A296068.

Cf. A001935, A299106, A299108.

nmax = 33; CoefficientList[Series[1/(1 - x Product[(1 + x^(2 k))/(1 - x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[1/(1 + (1 - x) QPochhammer[-1, x^2]/(2 QPochhammer[1/x, x^2])), {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[1/(1 - x EllipticTheta[2, 0, x]/(Sqrt[2] x^(1/8) EllipticTheta[2, Pi/4, Sqrt[x]])), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^(4*k))/(1 - x^k)).
G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k + x^(2*k) + x^(3*k))).
a(0) = 1; a(n) = Sum_{k=1..n} A001935(k-1)*a(n-k).

A307901 Expansion of 1/(1 - x * theta_4(x)), where theta_4() is the Jacobi theta function.

Original entry on oeis.org

1, 1, -1, -3, -1, 7, 11, -5, -33, -25, 53, 123, 9, -297, -363, 323, 1273, 657, -2415, -4407, 957, 12069, 11465, -16887, -47915, -12939, 104431, 152029, -85529, -476579, -333905, 803237, 1752799, 11597, -4349949, -5019855, 5068735, 18311655, 8392559, -35953969

Offset: 0

Ilya Gutkovskiy, May 04 2019

sign

Robert Israel, Table of n, a(n) for n = 0..5045
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A002448, A015128, A032803, A299108.

S:= series(1/(1-x*JacobiTheta4(0,x)),x,101):
seq(coeff(S,x,j),j=0..100);  # Robert Israel, Nov 03 2019

nmax = 39; CoefficientList[Series[1/(1 - x EllipticTheta[4, 0, x]), {x, 0, nmax}], x]
nmax = 39; CoefficientList[Series[1/(1 - x Product[(1 - x^k)/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: Sum_{k>=0} x^k * theta_4(x)^k.
G.f.: 1/(1 - x * Sum_{k=-oo..oo} (-1)^k * x^(k^2)).
G.f.: 1/(1 - x * Product_{k>=1} (1 - x^k)/(1 + x^k)).

A320654 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)/(1 - x^k)).

Original entry on oeis.org

1, 2, 8, 32, 126, 496, 1952, 7680, 30216, 118882, 467728, 1840224, 7240160, 28485616, 112073536, 440941056, 1734834302, 6825515600, 26854243752, 105655081568, 415688349456, 1635480294080, 6434618135968, 25316300481024, 99604212169632, 391881866363890, 1541816293103184

Offset: 0

Ilya Gutkovskiy, Oct 18 2018

nonn

Invert transform of A015128.

N. J. A. Sloane, Transforms

Cf. A002448, A015128, A055887, A299108, A304969.

a:=series(1/(2-mul((1+x^k)/(1-x^k),k=1..100)),x=0,27): seq(coeff(a,x,n),n=0..26); # Paolo P. Lava, Apr 02 2019

nmax = 26; CoefficientList[Series[1/(2 - Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 26; CoefficientList[Series[1/(2 - 1/EllipticTheta[4, 0, x]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[PartitionsP[k - j] PartitionsQ[j], {j, 0, k}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 26}]

G.f.: 1/(2 - 1/theta_4(x)), where theta_() is the Jacobi theta function.

a(0) = 1; a(n) = Sum_{k=1..n} A015128(k)*a(n-k).

cp's OEIS Frontend

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

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A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

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A302020 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^(2*k))/(1 - x^(2*k-1))).

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A307901 Expansion of 1/(1 - x * theta_4(x)), where theta_4() is the Jacobi theta function.

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A320654 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)/(1 - x^k)).

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Maple

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A302020 Expansion of 1/(1 - xProduct_{k>=1} (1 + x^(2k))/(1 - x^(2*k-1))).