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 1-6 of 6 results.

A383627 Expansion of 1/( Product_{k=0..2} (1 - (3*k+1) * x) )^(1/3).

Original entry on oeis.org

1, 4, 19, 100, 562, 3304, 20062, 124744, 789553, 5065444, 32840347, 214681636, 1412786872, 9348241504, 62138211112, 414627600736, 2775808278058, 18636412183336, 125436195473662, 846145250012776, 5719044971926972, 38723124875350960, 262609593669266404
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Comments

In general, if m > 0 and g.f. = 1/(Product_{k=0..m-1} (1 - (m*k+1)*x))^(1/m), then a(n) ~ (m*(m-1) + 1)^(n + 1 - 1/m) / (Gamma(1/m) * Gamma(m+1)^(1/m) * m^(1 - 2/m) * n^(1 - 1/m)). - Vaclav Kotesovec, Aug 18 2025

Crossrefs

Cf. A383628, A383629, A383630, A383631, A383632, A383633.
Cf. A016223, A370781, A383935.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/prod(k=0, 2, 1-(3*k+1)*x)^(1/3))

Formula

a(n) ~ 7^(n + 2/3) / (Gamma(1/3) * 2^(1/3) * 3^(2/3) * n^(2/3)). - Vaclav Kotesovec, May 12 2025
a(n) = Sum_{k=0..n} binomial(n,k) * A383935(k). - Seiichi Manyama, Aug 18 2025

A383628 Expansion of 1/( Product_{k=0..3} (1 - (4*k+1) * x) )^(1/4).

Original entry on oeis.org

1, 7, 59, 553, 5555, 58597, 640789, 7201383, 82659891, 964698805, 11408855809, 136374495803, 1644405320701, 19971195162107, 244004256374395, 2996243293813273, 36950056359522771, 457349452121086917, 5678884294812093329, 70710759962448700955, 882616583068179751945
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Crossrefs

Cf. A383627, A383629, A383630, A383631, A383632, A383633.
Cf. A383634.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/prod(k=0, 3, 1-(4*k+1)*x)^(1/4))

Formula

a(n) ~ 13^(n + 3/4) / (Gamma(1/4) * 2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 18 2025

A383629 Expansion of 1/( Product_{k=0..4} (1 - (5*k+1) * x) )^(1/5).

Original entry on oeis.org

1, 11, 146, 2156, 34166, 569426, 9854436, 175552696, 3199485331, 59384374841, 1118636310726, 21329345218236, 410804181673996, 7978922735099756, 156074211110053016, 3071360731347145776, 60752572593061028911, 1207041376109801598421, 24073933939936470329806
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Crossrefs

Cf. A383627, A383628, A383630, A383631, A383632, A383633.
Cf. A383635.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 4, 1-(5*k+1)*x)^(1/5))

Formula

a(n) ~ 3^(n + 3/5) * 7^(n + 4/5) / (Gamma(1/5) * 2^(3/5) * 5^(4/5) * n^(4/5)). - Vaclav Kotesovec, Aug 18 2025

A383631 Expansion of 1/( Product_{k=0..7} (1 - (8*k+1) * x) )^(1/8).

Original entry on oeis.org

1, 29, 1009, 39005, 1618849, 70741469, 3214527633, 150606953757, 7231305564225, 354221417305757, 17641204276036657, 890872808134921949, 45521466404971069921, 2349568589682742349405, 122328082368695017498321, 6416984703345086646305181, 338833672698752842286404737
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Crossrefs

Cf. A383627, A383628, A383629, A383630, A383632, A383633.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 7, 1-(8*k+1)*x)^(1/8))

Formula

a(n) ~ 3^(n + 5/8) * 19^(n + 7/8) / (Gamma(1/8) * 2^(25/8) * 5^(1/8) * 7^(1/8) * n^(7/8)). - Vaclav Kotesovec, Aug 18 2025

A383632 Expansion of 1/( Product_{k=0..8} (1 - (9*k+1) * x) )^(1/9).

Original entry on oeis.org

1, 37, 1639, 80623, 4257424, 236721412, 13688641144, 816291120808, 49895692924132, 3112177949225236, 197407027057353724, 12699858803178669148, 826900665838817386456, 54398158759680212197576, 3610650035912536155468808, 241521616482786052388206408, 16265890564063100473094045146
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Crossrefs

Cf. A383627, A383628, A383629, A383630, A383631, A383633.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 8, 1-(9*k+1)*x)^(1/9))

Formula

a(n) ~ 4480^(8/9) * 73^(n + 8/9) / (40320 * Gamma(1/9) * n^(8/9)). - Vaclav Kotesovec, Aug 18 2025

A383633 Expansion of 1/( Product_{k=0..10} (1 - (11*k+1) * x) )^(1/11).

Original entry on oeis.org

1, 56, 3741, 277256, 22052713, 1846878936, 160878051401, 14454374710216, 1331486959280259, 125190717874655720, 11973642784650273211, 1161838196321182959096, 114133506709827074843495, 11331528323810252967417064, 1135444330405820622163425351, 114694796036872449398436891896
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Comments

In general, if m > 0 and g.f. = 1/(Product_{k=0..m-1} (1 - (m*k+1)*x))^(1/m), then a(n) ~ (m*(m-1) + 1)^(n + 1 - 1/m) / (Gamma(1/m) * Gamma(m+1)^(1/m) * m^(1 - 2/m) * n^(1 - 1/m)). - Vaclav Kotesovec, Aug 18 2025

Crossrefs

Cf. A383627, A383628, A383629, A383630, A383631, A383632.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 10, 1-(11*k+1)*x)^(1/11))

Formula

a(n) ~ 3^(n + 6/11) * 37^(n + 10/11) / (Gamma(1/11) * 2^(8/11) * 5^(2/11) * 7^(1/11) * 11^(10/11) * n^(10/11)). - Vaclav Kotesovec, May 12 2025
Showing 1-6 of 6 results.

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