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.

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

A383937 Expansion of 1 / ( (1-3*x) * (1-6*x) )^(2/3).

Original entry on oeis.org

1, 6, 33, 180, 990, 5508, 30978, 175824, 1005345, 5782590, 33418737, 193876092, 1128297276, 6583492080, 38498441400, 225550220544, 1323563204394, 7777806812892, 45762197971050, 269545947941160, 1589219394582996, 9378142402189176, 55385341859409948
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2025

Keywords

Links

Crossrefs

Cf. A016137, A383935.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1 / ( (1-3*x) * (1-6*x) )^(2/3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025
  • Mathematica
    CoefficientList[Series[1/((1-3*x)*(1-6*x))^(2/3),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    a(n) = (-3)^n*sum(k=0, n, 2^k*binomial(-2/3, k)*binomial(-2/3, n-k));

Formula

G.f.: B(x)^(2/3), where B(x) is the g.f. of A016137.
a(n) = (-3)^n * Sum_{k=0..n} 2^k * binomial(-2/3,k) * binomial(-2/3,n-k).
a(n) ~ Gamma(1/3) * 2^(n - 1/3) * 3^(n + 1/2) / (Pi * n^(1/3)). - Vaclav Kotesovec, Aug 18 2025
D-finite with recurrence n*a(n) +3*(-3*n+1)*a(n-1) +6*(3*n-2)*a(n-2)=0. - R. J. Mathar, Aug 26 2025

A383936 Expansion of 1 / ( (1-6*x) * (1+3*x)^2 )^(1/3).

Original entry on oeis.org

1, 0, 9, 18, 162, 648, 4050, 20412, 117369, 639576, 3628233, 20360970, 115858512, 659144304, 3772679976, 21637128240, 124518397770, 718139766240, 4151542099770, 24045292507860, 139520453553468, 810834881716080, 4719151317787452, 27502373918849544
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2025

Keywords

Links

Crossrefs

Cf. A383935.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1 / ( (1-6*x) * (1+3*x)^2 )^(1/3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025
  • Mathematica
    CoefficientList[Series[1/((1-6*x)*(1+3*x)^2)^(1/3),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    a(n) = (-3)^n*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(-1/3, k)*binomial(-2/3, n-k));

Formula

a(n) = (-3)^n * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(-1/3,k) * binomial(-2/3,n-k).
a(n) ~ 2^(n + 2/3) * 3^(n - 2/3) / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Aug 18 2025
D-finite with recurrence n*a(n) +3*(-n+1)*a(n-1) +18*(-n+1)*a(n-2)=0. - R. J. Mathar, Aug 26 2025
Showing 1-3 of 3 results.

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