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 1-3 of 3 results.

A377233 Expansion of 1/(1 - 9*x/(1-x))^(2/3).

Original entry on oeis.org

1, 6, 51, 456, 4191, 39174, 370329, 3529284, 33838854, 325978044, 3152058630, 30572797920, 297294956070, 2897207397420, 28286321963370, 276611636831640, 2708781551458665, 26559205696513590, 260695647288540915, 2561413004129212440, 25188928968792165495
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A052268, A361375, A377234, A377235.

Programs

  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-2/3, k)*binomial(n-1, n-k));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..n-1} (2+k/n) * a(k).
a(n) = ((11*n-5)*a(n-1) - 10*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-2/3,k) * binomial(n-1,n-k).
a(n) ~ Gamma(1/3) * 3^(11/6) * 2^(n - 5/3) * 5^(n - 2/3) / (Pi * n^(1/3)). - Vaclav Kotesovec, Oct 21 2024
a(n) = 6*hypergeom([5/3, 1-n], [2], -9) for n > 0. - Stefano Spezia, May 04 2025

A377234 Expansion of 1/(1 - 9*x/(1-x))^(4/3).

Original entry on oeis.org

1, 12, 138, 1524, 16455, 175152, 1846164, 19320456, 201093843, 2084105820, 21524823858, 221678089716, 2277558628869, 23352604052952, 239024756624520, 2442818071519104, 24932208295715538, 254166614639215032, 2588333499216072516, 26333774228774140680, 267693203735009601870
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A052268, A361375, A377233, A377235.

Programs

  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-4/3, k)*binomial(n-1, n-k));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..n-1} (4-k/n) * a(k).
a(n) = ((11*n+1)*a(n-1) - 10*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-4/3,k) * binomial(n-1,n-k).
a(n) ~ 3^(11/3) * 10^(n - 4/3) * n^(1/3) / Gamma(1/3). - Vaclav Kotesovec, Oct 21 2024
a(n) = 12*hypergeom([7/3, 1-n], [2], -9) for n > 0. - Stefano Spezia, May 04 2025

A377261 Expansion of 1/(1 - 9*x*(1 + x))^(5/3).

Original entry on oeis.org

1, 15, 195, 2340, 26910, 301158, 3307590, 35830080, 384072975, 4082949585, 43113860361, 452742067440, 4732188244290, 49266375442110, 511157395433610, 5287689996408612, 54555878321808435, 561579617798527185, 5768783256563735265, 59149668761521664040, 605472238745163334116
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A180400, A376568, A377260.
Cf. A377235.

Programs

  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-5/3, k)*binomial(k, n-k));

Formula

a(n) = 3*((3*n+2)*a(n-1) + (3*n+4)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-5/3,k) * binomial(k,n-k).
a(n) ~ Gamma(1/3) * n^(2/3) * 3^(n + 3/2) * (3 + sqrt(13))^(n + 5/3) / (Pi * 13^(5/6) * 2^(n + 11/3)). - Vaclav Kotesovec, May 03 2025
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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