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

A377235 Expansion of 1/(1 - 9*x/(1-x))^(5/3).

Original entry on oeis.org

1, 15, 195, 2355, 27285, 307833, 3409485, 37253805, 402847620, 4320615390, 46032234486, 487743084150, 5144152999650, 54041442437850, 565803538944450, 5906360704312770, 61495776957754725, 638808193722602175, 6622218378818049075, 68522901145021162275, 707856527414874575805
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A052268, A361375, A377233, A377234.

Programs

  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-5/3, k)*binomial(n-1, n-k));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..n-1} (5-2*k/n) * a(k).
a(n) = ((11*n+4)*a(n-1) - 10*(n-2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-5/3,k) * binomial(n-1,n-k).
a(n) ~ Gamma(1/3) * 3^(29/6) * 2^(n - 11/3) * 5^(n - 5/3) * n^(2/3) / Pi. - Vaclav Kotesovec, Oct 21 2024
a(n) = 15*hypergeom([8/3, 1-n], [2], -9) for n > 0. - Stefano Spezia, May 04 2025

A377260 Expansion of 1/(1 - 9*x*(1 + x))^(4/3).

Original entry on oeis.org

1, 12, 138, 1512, 16191, 170856, 1785042, 18514548, 190978047, 1961435736, 20074741596, 204870399552, 2085761241018, 21191569851312, 214930928188116, 2176565295933000, 22012171108148025, 222351327936731700, 2243667436429422150, 22618648367553735000, 227826739721910301245
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A180400, A376568, A377261.
Cf. A377234.

Programs

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

Formula

a(n) = 3*((3*n+1)*a(n-1) + (3*n+2)*a(n-2))/n for n > 1.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-4/3,k) * binomial(k,n-k).
a(n) ~ n^(1/3) * 3^(n+1) * (3 + sqrt(13))^(n + 4/3) / (13^(2/3) * Gamma(1/3) * 2^(n + 4/3)). - 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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