A296068 -id:A296068 - cp's OEIS frontend

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

1, 1, 5, 22, 101, 481, 2330, 11425, 56549, 281911, 1413465, 7120136, 36006362, 182681916, 929461993, 4740491107, 24229115109, 124069449335, 636376573943, 3268955179686, 16814509004601, 86593280920756, 446437797872016, 2303948443259841, 11900990745759578, 61526182236027756

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Main diagonal of A296068.

Cf. A001935, A001936, A001937, A001939, A001940, A001941, A092877, A093160, A295831, A296043, A296045.

Table[SeriesCoefficient[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[((1 - x^(4 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(EllipticTheta[2, 0, x]/EllipticTheta[2, Pi/4, x^(1/2)]/(16 x)^(1/8))^n, {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 4}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} ((1 - x^(4*k))/(1 - x^k))^n.

a(n) ~ c * d^n / sqrt(n), where d = 5.2749356339591798618290252741994029798069148326559... and c = 0.2726256757090475625917361066565981461455343437... - Vaclav Kotesovec, Dec 05 2017

A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j-1))/(1 + x^(2j)))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, -1, 0, 1, 3, -1, 0, 0, 1, 4, 0, -2, 1, 0, 1, 5, 2, -5, 3, 0, 0, 1, 6, 5, -8, 3, 2, -1, 0, 1, 7, 9, -10, -1, 9, -4, -1, 0, 1, 8, 14, -10, -10, 20, -7, -4, 2, 0, 1, 9, 20, -7, -24, 31, -2, -15, 5, 1, 0, 1, 10, 27, 0, -42, 36, 20, -40, 9, 8, -2, 0, 1, 11, 35, 12, -62, 28, 65, -75, 3, 27, -8, -1, 0

Offset: 0

Ilya Gutkovskiy, Dec 04 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 + 8)*x^3 + (1/24)*k*(k^3 - 18*k^2 + 59*k - 18)*x^4 + (1/120)*k*(k^4 - 30*k^3 + 215*k^2 - 330*k + 144)*x^5 + ...
Square array begins:
1,  1,  1,  1,   1,   1,  ...
0,  1,  2,  3,   4,   5,  ...
0, -1, -1,  0,   2,   5,  ...
0,  0, -2, -5,  -8, -10,  ...
0,  1,  3,  3,  -1, -10,  ...
0,  0,  2,  9,  20,  31,  ...

G. C. Greubel, Rows n=0..100 of antidiagonals, flattened

Columns k=0..8 give A000007, A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845 (with offset 0).
Main diagonal gives A296043.
Cf. A296068.

Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i - 1))/(1 + x^(2 i)))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(1/8) EllipticTheta[2, 0, x^(1/2)]/EllipticTheta[2, 0, x])^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^(2*j-1))/(1 + x^(2*j)))^k.

G.f. of column k: (x^(1/8)*theta_2(sqrt(x))/theta_2(x))^k, where theta_() is the Jacobi theta function.

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).

cp's OEIS Frontend

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j-1))/(1 + x^(2j)))^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2*j-1))/(1 + x^(2*j)))^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j-1))/(1 + x^(2j)))^k.

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