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 1-4 of 4 results.

A296164 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^n.

Original entry on oeis.org

1, 1, 3, 10, 35, 131, 498, 1919, 7459, 29170, 114653, 452552, 1792754, 7124040, 28386081, 113372690, 453743907, 1819317153, 7306575042, 29386858821, 118348662525, 477188876405, 1926137365804, 7782398551661, 31472648050930, 127384123318906, 515978637418884
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 06 2017

Keywords

Links

Crossrefs

Cf. A003105, A058484, A058539, A103262, A255526, A296163.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(3 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[1/((1 - x^(6 k - 1)) (1 - x^(6 k - 5)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
(* Calculation of constants {d,c}: *) With[{k = 3}, {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/((1 - x^(6*k-1))*(1 - x^(6*k-5)))^n.
a(n) ~ c * d^n / sqrt(n), where d = 4.129321588075726742506... and c = 0.25764349816429874321... - Vaclav Kotesovec, May 18 2018

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

A304629 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(5*k)))^n.

Original entry on oeis.org

1, 1, 3, 13, 51, 201, 819, 3389, 14131, 59341, 250703, 1064207, 4535091, 19390229, 83139955, 357354213, 1539272499, 6642769925, 28714955571, 124312591469, 538895612751, 2338948779320, 10162837993377, 44202371860240, 192431323820851, 838442649862701, 3656031108325651
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Links

Crossrefs

Cf. A096938, A255526, A285291, A296163, A296164, A304628.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(5 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[1/(1 - x^k + x^(2 k) - x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
(* Calculation of constants {d,c}: *) With[{k = 5}, {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/(1 - x^k + x^(2*k) - x^(3*k) + x^(4*k))^n.
a(n) ~ c * d^n / sqrt(n), where d = 4.445766346387064439086120427... and c = 0.267035948020079842478289... - Vaclav Kotesovec, May 18 2018
Showing 1-4 of 4 results.

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