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 1-5 of 5 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

A383598 Expansion of 1/( (1-x^2)^2 * (1-x^2-9*x) )^(1/3).

Original entry on oeis.org

1, 3, 19, 132, 1000, 7884, 63802, 525666, 4388518, 37010220, 314633944, 2692239012, 23161121641, 200158043223, 1736461678195, 15114944308560, 131950690469920, 1154858014686960, 10130508263000440, 89045875688728440, 784127521246844872, 6916291864328172336
Offset: 0

Views

Author

Seiichi Manyama, May 01 2025

Keywords

Links

Crossrefs

Cf. A383597, A383599.
Cf. A376805.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/( (1-x^2)^2 * (1-x^2-9*x) )^(1/3))); // Vincenzo Librandi, May 04 2025
  • Mathematica
    Table[Sum[(-9)^(n-2*k)* Binomial[-1/3, n-2*k]* Binomial[n-k,k],{k,0,Floor[n/2]}],{n,0,22}] (* Vincenzo Librandi, May 04 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, (-9)^(n-2*k)*binomial(-1/3, n-2*k)*binomial(n-k, k));

Formula

a(n) = Sum_{k=0..floor(n/2)} (-9)^(n-2*k) * binomial(-1/3,n-2*k) * binomial(n-k,k).
a(n) ~ ((9 + sqrt(85))/2)^(n+1) / (Gamma(1/3) * 3^(4/3) * 85^(1/6) * n^(2/3)). - Vaclav Kotesovec, May 02 2025

A383599 Expansion of 1/( (1-x^3)^2 * (1-x^3-9*x) )^(1/3).

Original entry on oeis.org

1, 3, 18, 127, 951, 7425, 59473, 484902, 4005720, 33425587, 281152551, 2380227705, 20259341335, 173218395228, 1486747223136, 12803424371263, 110579924167533, 957494150283249, 8309596928695417, 72260720257071936, 629526082305028041, 5493357757059584986
Offset: 0

Views

Author

Seiichi Manyama, May 01 2025

Keywords

Links

Crossrefs

Cf. A383597, A383598.
Cf. A376806.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/( (1-x^3)^2 * (1-x^3-9*x) )^(1/3))); // Vincenzo Librandi, May 04 2025
  • Mathematica
    Table[Sum[(-9)^(n-3*k)* Binomial[-1/3, n-3*k]* Binomial[n-2*k,k],{k,0,Floor[n/3]}],{n,0,22}] (* Vincenzo Librandi, May 04 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, (-9)^(n-3*k)*binomial(-1/3, n-3*k)*binomial(n-2*k, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (-9)^(n-3*k) * binomial(-1/3,n-3*k) * binomial(n-2*k,k).

A383603 Expansion of 1/( (1-x)^2 * (1-x-9*x^2) )^(1/3).

Original entry on oeis.org

1, 1, 4, 7, 28, 67, 250, 703, 2497, 7648, 26488, 85036, 291337, 960769, 3280486, 10993165, 37541611, 127077160, 434707756, 1481346064, 5078811037, 17388735001, 59756049838, 205310507773, 707095964617, 2436104710774, 8406778618336, 29027513057326
Offset: 0

Views

Author

Seiichi Manyama, May 01 2025

Keywords

Links

Crossrefs

Cf. A383597, A383604.
Cf. A383605.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/( (1-x)^2 * (1-x-9*x^2) )^(1/3))); // Vincenzo Librandi, May 06 2025
  • Mathematica
    CoefficientList[Series[1/((1-x)^2*(1-x-9*x^2))^(1/3),{x,0,27}],x] (* Stefano Spezia, May 02 2025 *)
Table[Sum[(-9)^k*Binomial[-1/3,k]*Binomial[n-k,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 06 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, (-9)^k*binomial(-1/3, k)*binomial(n-k, k));

Formula

a(n) = Sum_{k=0..floor(n/2)} (-9)^k * binomial(-1/3,k) * binomial(n-k,k).
a(n) ~ ((1 + sqrt(37))/2)^(n + 5/3) / (Gamma(1/3) * 3^(4/3) * 37^(1/6) * n^(2/3)). - Vaclav Kotesovec, May 02 2025

A383604 Expansion of 1/( (1-x)^2 * (1-x-9*x^3) )^(1/3).

Original entry on oeis.org

1, 1, 1, 4, 7, 10, 31, 70, 127, 328, 799, 1666, 4000, 9817, 22078, 52060, 126727, 296101, 699601, 1691350, 4024450, 9574393, 23081776, 55394488, 132650923, 319807159, 770872429, 1855190146, 4479086230, 10825202521, 26145137668, 63241928080, 153144714331
Offset: 0

Views

Author

Seiichi Manyama, May 01 2025

Keywords

Links

Crossrefs

Cf. A383597, A383603.
Cf. A217615.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/( (1-x)^2 * (1-x-9*x^3) )^(1/3))); // Vincenzo Librandi, May 06 2025
  • Mathematica
    CoefficientList[Series[1/((1-x)^2*(1-x-9*x^3))^(1/3),{x,0,32}],x] (* Stefano Spezia, May 02 2025 *)
Table[Sum[(-9)^k*Binomial[-1/3,k]*Binomial[n-2*k,k],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 06 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, (-9)^k*binomial(-1/3, k)*binomial(n-2*k, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (-9)^k * binomial(-1/3,k) * binomial(n-2*k,k).
Showing 1-5 of 5 results.

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