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.

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A084259 a(n) = n! * (n+1)! * (n+2)! * (n+3).

Original entry on oeis.org

6, 48, 1440, 103680, 14515200, 3483648000, 1316818944000, 737418608640000, 584035538042880000, 630758381086310400000, 901984484953423872000000, 1666867328193927315456000000, 3900469547973789918167040000000
Offset: 0

Views

Author

Benoit Cloitre, Jun 21 2003

Keywords

Crossrefs

Cf. A000142, A010790, A176037.

Programs

  • Mathematica
    a[n_] := n! * (n+1)! * (n+2)! * (n+3); Array[a, 15, 0] (* Amiram Eldar, May 07 2025 *)

Extensions

Definition adapted to data by Georg Fischer, May 10 2021

A335997 Triangle read by rows: T(n,k) = Product_{i=n-k+1..n} i! for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 6, 12, 12, 1, 24, 144, 288, 288, 1, 120, 2880, 17280, 34560, 34560, 1, 720, 86400, 2073600, 12441600, 24883200, 24883200, 1, 5040, 3628800, 435456000, 10450944000, 62705664000, 125411328000, 125411328000
Offset: 0

Views

Author

Werner Schulte, Jul 08 2020

Keywords

Comments

Based on some integer sequence a(n), n>0, define triangular arrays A(a;n,k) by recurrence: A(a;0,0) = 1, and A(a;i,j) = 0 if j<0 or j>i, and A(a;n,k) = n! / (n-k)! * A(a;n-1,k) + a(n) * A(a;n-1,k-1) for 0<=k<=n. Then, Product_{i=1..n} (1 + (a(i) / i!) * x) = Sum_{k=0..n} A(a;n,k) / T(n,k) * x^k for n>=0 with empty product 1 (case n=0).
For the row reversed triangle R(n,k) = Product_{i=k+1..n} i! with empty product 1 (case k=n) the terms of the matrix inverse M are given by M(n,n) = 1 for n >= 0 and M(n,n-1) = -n! for n > 0 otherwise 0. - Werner Schulte, Oct 25 2022

Examples

			The triangle starts:
n\k :  0     1      2        3         4         5         6
============================================================
  0 :  1
  1 :  1     1
  2 :  1     2      2
  3 :  1     6     12       12
  4 :  1    24    144      288       288
  5 :  1   120   2880    17280     34560     34560
  6 :  1   720  86400  2073600  12441600  24883200  24883200
  etc.

Crossrefs

Cf. A000012 (col_0), A000142 (col_1), A010790 (col_2), A176037 (col_3), A000178 (main diagonal and first subdiagonal).
Row sums equal A051399(n+1).
Cf. A009963, A093002, A193520.

Programs

  • Mathematica
    T[n_, k_] := Product[i!, {i, n - k + 1, n}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 08 2020 *)

Formula

T(n,k) = T(n,1) * T(n-1,k-1) for 0 < k <= n.
T(2*n,n) = A093002(n+1) for n >= 0.
T(n,k)/T(k,k) = A009963(n,k) for 0 <= k <= n.
(Sum_{k=0..n} T(n,k) * T(n,n-k))/T(n,n) = A193520(n) for n >= 0.
Showing 1-2 of 2 results.

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

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