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.

A383601 Expansion of 1/( (1-x) * (1-10*x)^2 )^(1/3).

Original entry on oeis.org

1, 7, 58, 514, 4705, 43879, 414208, 3943492, 37782346, 363760390, 3515819020, 34088616940, 331383573010, 3228590970430, 31514912933800, 308126549765440, 3016908101224105, 29576113797737695, 290271761086278610, 2851684765215491050, 28040613734007656545
Offset: 0

Views

Author

Seiichi Manyama, May 01 2025

Keywords

Links

Crossrefs

Cf. A383605, A383606.
Cf. A004988, A377233, A383597.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/( (1-x) * (1-10*x)^2 )^(1/3))); // Vincenzo Librandi, May 05 2025
  • Mathematica
    Table[Sum[(-9)^k* Binomial[-2/3,k]* Binomial[n,k],{k,0,n}],{n,0,22}] (* Vincenzo Librandi, May 05 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-2/3, k)*binomial(n, k));

Formula

a(n) = Sum_{k=0..n} (-9)^k * binomial(-2/3,k) * binomial(n,k).
n*a(n) = (11*n-4)*a(n-1) - 10*(n-1)*a(n-2) for n > 1.
a(n) ~ Gamma(1/3) * 2^(n - 2/3) * 5^(n + 1/3) / (Pi * 3^(1/6) * n^(1/3)). - Vaclav Kotesovec, May 02 2025
a(n) = hypergeom([2/3, -n], [1], -9). - 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

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

A376568 Expansion of 1/(1 - 9*x*(1 + x))^(2/3).

Original entry on oeis.org

1, 6, 51, 450, 4095, 37908, 354978, 3351348, 31833945, 303822090, 2910657321, 27970777926, 269484894081, 2602002636540, 25170322256010, 243876058527132, 2366251795228437, 22987502934573762, 223563791480714685, 2176402892261301990, 21206170582394740371
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2024

Keywords

Crossrefs

Cf. A180400, A377260, A377261.
Cf. A377233.

Programs

  • PARI
    a(n) = sum(k=0, n, (-9)^k*binomial(-2/3, 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(-2/3,k) * binomial(k,n-k).
a(n) ~ (3 + sqrt(13))^(n + 2/3) * 3^n / (Gamma(2/3) * 13^(1/3) * n^(1/3) * 2^(n + 2/3)). - Vaclav Kotesovec, Oct 26 2024
Showing 1-4 of 4 results.

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