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.

A384165 a(n) = Product_{k=0..n-1} (3*n+2*k).

Original entry on oeis.org

1, 3, 48, 1287, 48384, 2340135, 138378240, 9672183675, 780151357440, 71322093677835, 7287813911347200, 823100991923184975, 101819334240239616000, 13690816766440373134575, 1988199345147516813312000, 310120801435080997013527875, 51709528644340997758648320000
Offset: 0

Views

Author

Seiichi Manyama, May 21 2025

Keywords

Links

Crossrefs

Cf. A384164, A384166.

Programs

  • Magma
    [1] cat  [&*[(3*n + 2*k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025
  • Mathematica
    a[n_]:=Product[(3*n+2*k),{k,0,n-1}]; Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 22 2025 *)
  • PARI
    a(n) = prod(k=0, n-1, 3*n+2*k);
  • Python
    from math import prod
def A384165(n): return prod(3*n+i for i in range(0,n<<1,2)) # Chai Wah Wu, May 21 2025
  • Sage
    def a(n): return 2^n*rising_factorial(3*n/2, n)

Formula

a(n) = 2^n * RisingFactorial(3*n/2,n).
a(n) = n! * [x^n] 1/(1 - 2*x)^(3*n/2).
a(n) = (3/5) * 2^n * n! * binomial(5*n/2,n) for n > 0.

A384241 a(n) = Product_{k=0..n-1} (3*n-4*k).

Original entry on oeis.org

1, 3, 12, 45, 0, -3465, -60480, -626535, 0, 204417675, 6227020800, 104928949125, 0, -77849405258625, -3379030566912000, -78792721832199375, 0, 104312208642352585875, 5875458349746585600000, 174954117301479619228125, 0, -362526128354588965187045625, -25100240092118201519308800000
Offset: 0

Views

Author

Seiichi Manyama, May 22 2025

Keywords

Crossrefs

Cf. A064352, A343445.
Cf. A383996, A384166.

Programs

  • PARI
    a(n) = prod(k=0, n-1, 3*n-4*k);
  • Sage
    def a(n): return 4^n*falling_factorial(3*n/4, n)

Formula

a(n) = 4^n * FallingFactorial(3*n/4,n).
a(n) = n! * [x^n] (1 + 4*x)^(3*n/4).
a(n) = 3 * (-1)^(n-1) * A383996(n) for n > 0.
a(4*n) = 0 for n > 0.

A384172 a(n) = 4^n * n! * binomial(7*n/4,n) * Sum_{k=1..n} 1/(3*n+4*k).

Original entry on oeis.org

1, 24, 851, 40832, 2483269, 183241728, 15912395295, 1590131687424, 179766351690345, 22685041361848320, 3161081216499580395, 482101740659382681600, 79876921394710650447405, 14287114673531430042009600, 2743817201103924825303993975, 563131793021994402478188134400
Offset: 1

Views

Author

Seiichi Manyama, May 21 2025

Keywords

Crossrefs

Cf. A384137, A384171.
Cf. A384166.

Programs

  • PARI
    a(n) = sum(k=0, n, k*(3*n+4)^(k-1)*4^(n-k)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} k * (3*n+4)^(k-1) * 4^(n-k) * |Stirling1(n,k)|.
a(n) = n! * [x^n] ( -log(1 - 4*x)/(4 * (1 - 4*x)^(3*n/4+1)) ).
Showing 1-3 of 3 results.

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

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