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.

A137268 Triangle T(n, k) = k! * (k+1)^(n-k), read by rows.

Original entry on oeis.org

1, 2, 2, 4, 6, 6, 8, 18, 24, 24, 16, 54, 96, 120, 120, 32, 162, 384, 600, 720, 720, 64, 486, 1536, 3000, 4320, 5040, 5040, 128, 1458, 6144, 15000, 25920, 35280, 40320, 40320, 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880
Offset: 1

Views

Author

Roger L. Bagula, Mar 12 2008

Keywords

Comments

Essentially the same as A104001.

Examples

			Triangle begins as:
    1;
    2,     2;
    4,     6,     6;
    8,    18,    24,     24;
   16,    54,    96,    120,    120;
   32,   162,   384,    600,    720,     720;
   64,   486,  1536,   3000,   4320,    5040,    5040;
  128,  1458,  6144,  15000,  25920,   35280,   40320,   40320;
  256,  4374, 24576,  75000, 155520,  246960,  322560,  362880,  362880;
  512, 13122, 98304, 375000, 933120, 1728720, 2580480, 3265920, 3628800, 3628800;

Links

Crossrefs

Cf. A061711, A066319, A074143, A104001, A152684.

Programs

  • Magma
    [Factorial(k)*(k+1)^(n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 28 2022
  • Mathematica
    T[n_, k_]:= k!*(k+1)^(n-k);
Table[T[n, k], {n, 12}, {k, n}]//Flatten
  • SageMath
    def A137268(n,k): return factorial(k)*(k+1)^(n-k)
flatten([[A137268(n,k) for k in range(1,n+1)] for n in range(14)]) # G. C. Greubel, Nov 28 2022

Formula

J(b, n) = (b+1)^(n-b)*b! if n > b, otherwise n! (notation of Chung and Graham).
From G. C. Greubel, Nov 28 2022: (Start)
T(n, k) = k! * (k+1)^(n-k).
T(n, n-2) = 2*A074143(n), n > 1.
T(2*n, n) = A152684(n).
T(2*n, n-1) = A061711(n).
T(2*n+1, n+1) = A066319(n). (End)

Extensions

Edited by G. C. Greubel, Nov 28 2022

A104001 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns starting with fixed m, 2

Original entry on oeis.org

2, 4, 6, 8, 18, 24, 16, 54, 96, 120, 32, 162, 384, 600, 720, 64, 486, 1536, 3000, 4320, 5040, 128, 1458, 6144, 15000, 25920, 35280, 40320, 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 512, 13122, 98304, 375000, 933120, 1728720, 2580480, 3265920, 3628800
Offset: 3

Views

Author

Ralf Stephan, Feb 26 2005

Keywords

Examples

			Triangle begins as:
    2;
    4,    6;
    8,   18,   24;
   16,   54,   96,   120;
   32,  162,  384,   600,   720;
   64,  486, 1536,  3000,  4320,  5040;
  128, 1458, 6144, 15000, 25920, 35280, 40320;

Links

Crossrefs

Cf. A000079, A000244, A000302, A137268,

Programs

  • Magma
    [Factorial(k-1)*(k-1)^(n-k): k in [3..n], n in [3..15]]; // G. C. Greubel, Nov 29 2022
  • Mathematica
    Table[(k-1)!*(k-1)^(n-k), {n,3,15}, {k,3,n}]//Flatten (* G. C. Greubel, Nov 29 2022 *)
  • SageMath
    def A104001(n,k): return factorial(k-1)*(k-1)^(n-k)
flatten([[A104001(n,k) for k in range(3,n+1)] for n in range(3,16)]) # G. C. Greubel, Nov 29 2022

Formula

T(n, k) = (k-2)! * (k-1)^(n+1-k).
From G. C. Greubel, Nov 29 2022: (Start)
T(n, 3) = A000079(n-2).
T(n, 4) = 6*A000244(n-4).
T(n, 5) = 4!*A000302(n-5).
T(2*n-3, n) = A152684(n-1). (End)

A355006 Triangle read by rows. T(n, k) = n^(n - k) * |Stirling1(n, k)|.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 18, 9, 1, 0, 384, 176, 24, 1, 0, 15000, 6250, 875, 50, 1, 0, 933120, 355104, 48600, 3060, 90, 1, 0, 84707280, 29647548, 3899224, 252105, 8575, 147, 1, 0, 10569646080, 3425697792, 430309376, 27725824, 1003520, 20608, 224, 1
Offset: 0

Views

Author

Peter Luschny, Jun 17 2022

Keywords

Examples

			Table T(n, k) begins:
[0] 1;
[1] 0,           1;
[2] 0,           2,          1;
[3] 0,          18,          9,         1;
[4] 0,         384,        176,        24,        1;
[5] 0,       15000,       6250,       875,       50,       1;
[6] 0,      933120,     355104,     48600,     3060,      90,     1;
[7] 0,    84707280,   29647548,   3899224,   252105,    8575,   147,   1;
[8] 0, 10569646080, 3425697792, 430309376, 27725824, 1003520, 20608, 224, 1;

Crossrefs

A152684 (column 1), A006002 (subdiagonal), A092985 (row sums), A355007.

Programs

  • Maple
    seq(seq(n^(n - k)*abs(Stirling1(n, k)), k = 0..n), n = 0..9);
  • Mathematica
    T[n_, k_] := If[n == k == 0, 1, n^(n - k) * Abs[StirlingS1[n, k]]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 17 2022 *)
Showing 1-3 of 3 results.

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

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