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.

A023991 Sum of exponents of primes in multinomial coefficient M(3n; n+1,n,n-1).

Original entry on oeis.org

1, 4, 6, 8, 8, 12, 12, 13, 13, 15, 17, 20, 19, 22, 21, 22, 21, 24, 25, 26, 25, 31, 30, 32, 30, 31, 33, 33, 32, 36, 34, 36, 34, 36, 36, 37, 36, 40, 40, 42, 40, 45, 48, 49, 49, 51, 50, 52, 49, 50, 50, 53, 50, 56, 53, 53, 53, 55, 58, 60, 59, 62, 60, 60, 55, 58, 59, 61, 60, 65, 62, 65, 63, 66, 69, 68
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A022559, A023978, A076191, A248707.
Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Multinomial[n+1, n, n-1]]; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega((3*n)! / ((n-1)!*n!*(n+1)!)); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A248707(n)).
a(n) = A022559(3*n) - A022559(n-1) - A022559(n) - A022559(n+1) = A022559(3*n) - 3*A022559(n) - A001222(n+1) + A001222(n) = A023978(n) - A076191(n). (End)

A248708 a(n) = f(4*n+2)/(f(n-1)*f(n)*f(n+1)*f(n+2)), where f(k) = k!.

Original entry on oeis.org

60, 12600, 2522520, 514594080, 107550162720, 22969641895200, 4995897112206000, 1103284402265073600, 246784661070292144800, 55803246694136969227200, 12736017918577260592084800, 2930174751896446667689440000, 678879630375093886522676256000, 158257286142440155623613107216000
Offset: 1

Views

Author

Clark Kimberling, Oct 12 2014

Keywords

Comments

These are multinomial coefficients.

Examples

			a(2) = 10!/(1!*2!*3!*4!) = 12600.

Links

Crossrefs

Cf. A000142, A001700, A248707, A248709, A248710.

Programs

  • Mathematica
    Table[(4 n + 2)!/((n - 1)! n! (n + 1)! (n + 2)!), {n, 1, 20}]
  • Sage
    [factorial(4*n + 2)/(factorial(n - 1)* factorial(n)*factorial(n + 1)*factorial(n + 2)) for n in range(1,14)] # Stefano Spezia, Aug 16 2024

Formula

a(n) ~ 2^(8*n+7/2) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Oct 19 2014

A248709 a(n) = f(5*n)/(f(n-2)*f(n-1)*f(n)*f(n+1)*f(n+2)), where f(k) = k!.

Original entry on oeis.org

12600, 37837800, 97772875200, 247365374256000, 629483036137956000, 1621828071329658192000, 4234824783966213204768000, 11198994141198650820008976000, 29959571750765218953790679280000, 80980722442318386832096206093840000, 220917676017677910841226480887103040000
Offset: 2

Views

Author

Clark Kimberling, Oct 12 2014

Keywords

Comments

These are multinomial coefficients.

Examples

			a(3) = 15!/(1!*2!*3!*4!*5!) = 37837800.

Links

Crossrefs

Cf. A000142, A001700, A248707, A248708, A248710.

Programs

  • Mathematica
    Table[(5 n)!/((n - 2)! (n - 1)! n! (n + 1)! (n + 2)!), {n, 2, 20}]
  • Sage
    [factorial(5*n)/(factorial(n - 2)*factorial(n - 1)*factorial(n)*factorial(n + 1)*factorial(n + 2)) for n in range(2,14)] # Stefano Spezia, Aug 16 2024

Formula

a(n) ~ 5^(5*n+1/2) / (4*Pi^2*n^2). - Vaclav Kotesovec, Oct 19 2014

A248710 a(n) = f(6*n+3)/(f(n-2)*f(n-1)*f(n)*f(n+1)*f(n+2)*f(n+3)), where f(k) = k!.

Original entry on oeis.org

37837800, 2053230379200, 86825246363856000, 3434459445168687936000, 133396980694935715950192000, 5173935293233776678844146912000, 201687837026151453996918852912960000, 7920886423528046052820994110450678080000, 313629810506083768747620025974652020366480000
Offset: 2

Views

Author

Clark Kimberling, Oct 12 2014

Keywords

Comments

These are multinomial coefficients.

Examples

			a(3) = 21!/(1!*2!*3!*4!*5!*6!) = 2053230379200.

Links

Crossrefs

Cf. A000142, A001700, A248707, A248708, A248709.

Programs

  • Mathematica
    Table[(6 n + 3)!/((n - 2)! (n - 1)! n! (n + 1)! (n + 2)! (n + 3)!), {n, 2, 20}]
Table[(6n+3)!/Times@@((n+Range[-2,3])!)  ,{n,2,20}] (* Harvey P. Dale, Jul 20 2020 *)
  • Sage
    [factorial(6*n + 3)/(factorial(n - 2)*factorial(n - 1)*factorial(n)*factorial(n + 1)*factorial(n + 2)*factorial(n + 3)) for n in range(2,11)] # Stefano Spezia, Aug 16 2024

Formula

a(n) ~ 3^(5/2) * 6^(6*n+1) / (Pi^(5/2)*n^(5/2)). - Vaclav Kotesovec, Oct 19 2014
Showing 1-4 of 4 results.

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