cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A296043 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^n.

Original entry on oeis.org

1, 1, -1, -5, -1, 31, 65, -90, -641, -644, 3329, 11386, -1471, -87021, -164634, 317935, 1881471, 1418719, -11370760, -33937951, 17468929, 294971868, 468897758, -1304743033, -6275603903, -2804572819, 42665919997, 109181454826, -106020803386, -1063546684834, -1362993953395
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2017

Keywords

Links

Crossrefs

Main diagonal of A296067.
Cf. A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845, A296044, A296045.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 + x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[(x^(1/8) EllipticTheta[2, 0, x^(1/2)]/EllipticTheta[2, 0, x])^n, {x, 0, n}], {n, 0, 30}]

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^n.

Original entry on oeis.org

1, 1, 3, 13, 55, 231, 981, 4222, 18351, 80320, 353453, 1562364, 6932185, 30856541, 137725710, 616190583, 2762605791, 12408541299, 55825435656, 251523510045, 1134741006825, 5125453110196, 23175983361270, 104899547541255, 475228898015025, 2154737528486881, 9777332125043577
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2017

Keywords

Links

Crossrefs

Cf. A006950, A106337, A192540, A273225, A273226, A273228, A295832, A296043, A296044.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(2 (-x)^(1/8)/EllipticTheta[2, 0, Sqrt[-x]])^n, {x, 0, n}], {n, 0, 26}]
Table[(-1)^n * 2^n * SeriesCoefficient[1/(QPochhammer[-1, x]*QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 07 2020 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, 4/Sqrt[Pi*(77/2 - 4*s*(-r*s)^(7/8) * Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[-r*s]])]} /. FindRoot[{s == (2*(-r*s)^(1/8))/EllipticTheta[2, 0, Sqrt[-r*s]], 7*I*r + 2*(-r*s)^(7/8)*Sqrt[r*s] * Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[-r*s]] == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

Formula

a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 - x^(4*k)))^n.
a(n) ~ c * d^n / sqrt(n), where d = 4.62579056836776492108784045382518984897... (see A192540) and c = 0.255113338880004277664416308115912337... - Vaclav Kotesovec, Dec 05 2017

A296163 a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 1, 5, 22, 105, 501, 2456, 12160, 60801, 306130, 1550255, 7887034, 40281720, 206405967, 1060602800, 5463059772, 28199365873, 145832364580, 755420838614, 3918935839970, 20357605331355, 105878815699042, 551273881133750, 2873161931172668, 14988243880188600
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 06 2017

Keywords

Links

Crossrefs

Cf. A008485, A035959, A121591, A263002, A270913, A285928, A296044, A296162.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 - x^(5 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k) + x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
(* Calculation of constant d: *) With[{k = 5}, 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 *)

Formula

a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k) + x^(3*k) + x^(4*k))^n.
a(n) ~ c * d^n / sqrt(n), where d = 5.3271035802753567624196808294779171420899175782347488197... and c = 0.2712048688090020853684153670711011713396954... - Vaclav Kotesovec, May 13 2018

A296068 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 - x^(2*j-1)))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 4, 0, 1, 5, 14, 22, 18, 6, 0, 1, 6, 20, 40, 48, 32, 9, 0, 1, 7, 27, 65, 101, 99, 55, 12, 0, 1, 8, 35, 98, 185, 236, 194, 90, 16, 0, 1, 9, 44, 140, 309, 481, 518, 363, 144, 22, 0, 1, 10, 54, 192, 483, 882, 1165, 1080, 657, 226, 29, 0, 1, 11, 65, 255, 718, 1498, 2330, 2665, 2162, 1155, 346, 38, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 04 2017

Keywords

Examples

			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,  2,   5,   9,   14,   20,  ...
0,  3,  10,  22,   40,   65,  ...
0,  4,  18,  48,  101,  185,  ...
0,  6,  32,  99,  236,  481,  ...

Crossrefs

Columns k=0..8 give A000007, A001935, A001936, A001937, A093160, A001939, A001940, A001941, A092877.
Main diagonal gives A296044.
Antidiagonal sums give A302020.
Cf. A296067.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i))/(1 - x^(2 i - 1)))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[((1 - x^(4 i))/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[2^(-k/2) (EllipticTheta[2, 0, x]/(x^(1/8) EllipticTheta[2, Pi/4, Sqrt[x]]))^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

Formula

G.f. of column k: Product_{j>=1} ((1 + x^(2*j))/(1 - x^(2*j-1)))^k.
G.f. of column k: Product_{j>=1} ((1 - x^(4*j))/(1 - x^j))^k.
G.f. of column k: 2^(-k/2)*(theta_2(0,x)/(x^(1/8)*theta_2(Pi/4,sqrt(x))))^k, where theta_() is the Jacobi theta function.

A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 0, 3, 19, 101, 501, 2486, 12398, 62329, 315436, 1605330, 8207552, 42124368, 216903051, 1119974861, 5796944342, 30068145889, 156250892593, 813310723907, 4239676354631, 22130265931880, 115654632452514, 605081974091853, 3168828466966365, 16610409114771876, 87141919856550506
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Comments

Number of partitions of n into 2 or more parts of n kinds. - Ilya Gutkovskiy, May 16 2018

Links

Crossrefs

Cf. A000065, A008485, A022567, A093160, A270913, A285927, A285928, A286653, A296044, A296162, A296163, A304626.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 - x^(n k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 25}]

Formula

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724... and c = 0.268015212710733315686... - Vaclav Kotesovec, May 16 2018

A296162 a(n) = [x^n] Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 1, 5, 19, 89, 406, 1913, 9073, 43505, 209971, 1019390, 4971781, 24343037, 119579006, 589050663, 2908727839, 14393759457, 71360342129, 354372852011, 1762416036422, 8776797353574, 43761058841638, 218431753457637, 1091385769314272, 5458068218285909, 27319061357620056
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 06 2017

Keywords

Links

Crossrefs

Cf. A000726, A008485, A121596, A128758, A270913, A285927, A296044, A296163.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 - x^(3 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 3}, 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 *)

Formula

a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k))^n.
a(n) ~ c * d^n / sqrt(n), where d = 5.1069752682291604222843644516987970799026747758649349... and c = 0.271879273312907861082536692355942116774864... - Vaclav Kotesovec, May 13 2018

A304628 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(4*k)))^n.

Original entry on oeis.org

1, 1, 3, 13, 47, 181, 729, 2948, 12031, 49540, 205153, 853546, 3565505, 14943839, 62810786, 264650683, 1117486463, 4727486583, 20032950744, 85017558081, 361289789377, 1537198394570, 6547611493822, 27917246924099, 119141276756545, 508884954441331, 2175284934712217, 9305217981192748
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Links

Crossrefs

Cf. A070048, A255526, A285290, A296044, A296164, A304629.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]
Table[SeriesCoefficient[Product[((1 - x^(8 k - 4))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]
(* Calculation of constants {d,c}: *) With[{k = 4}, {1/r, Sqrt[QPochhammer[-1, (r*s)^k] / (2*Pi*(r^2*s*Derivative[0, 2][QPochhammer][-1, r*s] - k^2*(r*s)^(2*k) * Derivative[0, 2][QPochhammer][-1, (r*s)^k] - k*(1 + k)*(r*s)^k * Derivative[0, 1][QPochhammer][-1, (r*s)^k]))]} /. FindRoot[{s == QPochhammer[-1, r*s]/QPochhammer[-1, (r*s)^k], QPochhammer[-1, (r*s)^k] + k*(r*s)^k*Derivative[0, 1][QPochhammer][-1, (r*s)^k] == r*Derivative[0, 1][QPochhammer][-1, r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

Formula

a(n) = [x^n] Product_{k>=1} ((1 - x^(8*k-4))/(1 - x^(2*k-1)))^n.
a(n) ~ c * d^n / sqrt(n), where d = 4.3582188263213968630940316689... and c = 0.266443662680498334500839... - Vaclav Kotesovec, May 18 2018

A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)*(1 - x^(2*k)))^n.

Original entry on oeis.org

1, -1, -3, 14, -11, -81, 282, -57, -2043, 5405, 2417, -46476, 94522, 110512, -943407, 1505289, 2807589, -16888311, 23645199, 46006542, -265972791, 472882620, 187884672, -3981273597, 14234579226, -19187383356, -78662039004, 502118911904, -847583768679, -2627514175002
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 19 2018

Keywords

Crossrefs

Cf. A002171, A002288, A008705, A030204, A030211, A066535, A296043, A296044, A319457.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - 4 DivisorSigma[1, k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]

Formula

a(n) = [x^n] Product_{k>=1} (1 - x^(2*k))^(2*n)/(1 + x^k)^n.
a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k).

A319457 a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)*(1 - x^(2*k)))^n.

Original entry on oeis.org

1, 1, 7, 31, 175, 931, 5209, 29114, 165087, 940828, 5396777, 31090962, 179832625, 1043516371, 6072302726, 35420582431, 207051636799, 1212583329959, 7113193757656, 41788933655049, 245831162935825, 1447891754747672, 8537111315442222, 50387162650271055, 297664212003582753
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 19 2018

Keywords

Crossrefs

Cf. A002513, A008485, A270919, A296043, A296044, A319455, A319456.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[1/(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Exp[n Sum[(4 DivisorSigma[1, k] - DivisorSigma[1, 2 k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 24}]

Formula

a(n) = [x^n] Product_{k>=1} (1 + x^k)^n/(1 - x^(2*k))^(2*n).
a(n) = [x^n] exp(n*Sum_{k>=1} (4*sigma(k) - sigma(2*k))*x^k/k).
Showing 1-9 of 9 results.

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