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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A268363 Number of n X 2 arrays containing 2 copies of 0..n-1 with row sums equal.

Original entry on oeis.org

1, 1, 4, 12, 96, 480, 5760, 40320, 645120, 5806080, 116121600, 1277337600, 30656102400, 398529331200, 11158821273600, 167382319104000, 5356234211328000, 91055981592576000, 3278015337332736000, 62282291409321984000, 2491291656372879360000
Offset: 0

Views

Author

R. H. Hardin, Feb 03 2016

Keywords

Comments

Column 2 of A268367.

Examples

			Some solutions for n=5
..3..1....1..3....4..0....0..4....0..4....4..0....4..0....4..0....2..2....1..3
..1..3....2..2....3..1....3..1....2..2....3..1....1..3....2..2....0..4....0..4
..0..4....0..4....0..4....2..2....1..3....1..3....0..4....1..3....3..1....4..0
..0..4....0..4....2..2....1..3....4..0....0..4....2..2....0..4....4..0....3..1
..2..2....3..1....1..3....0..4....1..3....2..2....1..3....3..1....3..1....2..2

Links

Crossrefs

Cf. A065140, A121627, A268367.
Cf. A158867.

Programs

  • Magma
    [2^Floor(n/2)*Factorial(n): n in [0..25]]; // G. C. Greubel, Mar 08 2022
  • Mathematica
    Table[2^Floor[n/2] n!, {n,0,25}] (* Michael De Vlieger, Aug 11 2016 *)
  • Sage
    [2^(n//2)*factorial(n) for n in (0..25)] # G. C. Greubel, Mar 08 2022

Formula

a(n) = 2^floor(n/2) * n!. - Joel B. Lewis, Aug 11 2016
a(2n) = A065140(n) for terms > a(1) - Terry D. Grant, May 28 2017
a(n) = A158867(n, n) for n > 0. - G. C. Greubel, Mar 08 2022

Extensions

Title clarified by Joel B. Lewis, Aug 11 2016
a(0)=1 prepended by Alois P. Heinz, May 27 2017

A158868 Triangle T(n, k) = (2*n+1)!! * 2^(floor((n-1)/2) + floor(k/2) + 1) * Beta(floor(n/2) + floor((k-1)/2) + 2, floor((n-1)/2) + floor(k/2) + 2), read by rows.

Original entry on oeis.org

1, 5, 2, 14, 7, 6, 126, 54, 54, 24, 594, 297, 264, 132, 120, 7722, 3432, 3432, 1560, 1560, 720, 51480, 25740, 23400, 11700, 10800, 5400, 5040, 875160, 397800, 397800, 183600, 183600, 85680, 85680, 40320, 7558200, 3779100, 3488400, 1744200, 1627920, 813960, 766080, 383040, 362880
Offset: 1

Views

Author

Roger L. Bagula, Mar 28 2009

Keywords

Examples

			Triangle begins as:
        1;
        5,       2;
       14,       7,       6;
      126,      54,      54,      24;
      594,     297,     264,     132,     120;
     7722,    3432,    3432,    1560,    1560,    720;
    51480,   25740,   23400,   11700,   10800,   5400,   5040;
   875160,  397800,  397800,  183600,  183600,  85680,  85680,  40320;
  7558200, 3779100, 3488400, 1744200, 1627920, 813960, 766080, 383040, 362880;

Links

Crossrefs

Cf. A158867.

Programs

  • Mathematica
    T[n_, k_]:= (2*n+1)!!*2^(1+Floor[n/2] +Floor[(k-1)/2])*Beta[Floor[n/2] +Floor[(k- 1)/2] +2, Floor[(n-1)/2] +Floor[k/2] +2];
Table[T[n, k], {n,10}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 07 2022 *)
  • Sage
    def T(n,k): return (2*n+1).multifactorial(2)*2^(1+(n//2)+((k-1)//2))*beta(2+(n//2)+((k-1)//2), 2+((n-1)//2)+(k//2))
flatten([[T(n,k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Mar 07 2022

Formula

T(n, k) = (2*n+1)!! * 2^(floor((n-1)/2) + floor(k/2) + 1) * Beta(floor(n/2) + floor((k-1)/2) + 2, floor((n-1)/2) + floor(k/2) + 2).
T(n, n) = n!. - G. C. Greubel, Mar 07 2022

Extensions

Edited by G. C. Greubel, Mar 07 2022
Showing 1-2 of 2 results.

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