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 1-2 of 2 results.

A384300 a(n) = Product_{k=0..2*n-1} (3*n+k-2).

Original entry on oeis.org

1, 2, 840, 665280, 980179200, 2346549004800, 8326896754176000, 41098950018846720000, 269397128065642536960000, 2264501147602213494374400000, 23751156416080627455365283840000, 304080322557324667642345606348800000, 4667216066941750219330172809445376000000
Offset: 0

Views

Author

Seiichi Manyama, May 25 2025

Keywords

Crossrefs

Cf. A384301, A384302, A384303.
Cf. A384262.

Programs

  • PARI
    a(n) = (2*n)!*binomial(5*n-3, 2*n);

Formula

a(n) = RisingFactorial(3*n-2,2*n).
a(n) = (2*n)! * [x^(2*n)] 1/(1 - x)^(3*n-2).
a(n) = (2*n)! * binomial(5*n-3,2*n).
D-finite with recurrence 3*(3*n-4)*(3*n-5)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-7)*(5*n-6)*a(n-1)=0. - R. J. Mathar, May 26 2025

A384263 a(n) = Product_{k=0..n-1} (3*n+k-1).

Original entry on oeis.org

1, 2, 30, 720, 24024, 1028160, 53721360, 3315312000, 235989936000, 19033511777280, 1715456253772800, 170866312333516800, 18638248113733248000, 2209723830420986880000, 282926061171849199104000, 38906746608339829739520000, 5719086709283091520696320000, 894889312443445445244518400000
Offset: 0

Views

Author

Seiichi Manyama, May 23 2025

Keywords

Crossrefs

Cf. A061924, A384164, A384262.

Programs

  • Mathematica
    a[n_]:=n!*Binomial[4*n-2,n]; Array[a,18,0] (* Stefano Spezia, Sep 04 2025 *)
  • PARI
    a(n) = prod(k=0, n-1, 3*n+k-1);
  • Python
    from sympy import rf
def a(n): return rf(3*n-1, n)
  • Sage
    def a(n): return rising_factorial(3*n-1, n)

Formula

a(n) = RisingFactorial(3*n-1,n).
a(n) = n! * [x^n] 1/(1 - x)^(3*n-1).
a(n) = n! * binomial(4*n-2,n).
From Stefano Spezia, Sep 04 2025: (Start)
E.g.f.: (1 + 3*hypergeom([-1/4, 1/4, 1/2], [-1/3, 1/3], 2^8*x/3^3])/4.
a(n) ~ 2^(8*n-7)*3^(-3*n-1/2)*exp(-n)*n^(n-1)*(144*n - 13). (End)
Showing 1-2 of 2 results.

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

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