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 1-2 of 2 results.

A126596 a(n) = binomial(4*n,n)*(2*n+1)/(3*n+1).

Original entry on oeis.org

1, 3, 20, 154, 1260, 10659, 92092, 807300, 7152444, 63882940, 574221648, 5188082354, 47073334100, 428634152730, 3914819231400, 35848190542920, 329007937216860, 3025582795190340, 27872496751392496, 257172019222240200, 2376196095585231920, 21983235825545286435
Offset: 0

Views

Author

Philippe Deléham, Mar 13 2007

Keywords

Comments

Number of standard Young tableaux of shape [3n,n]. Also the number of binary words with 3n 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 3 words are: 1011, 1101, 1110. - Alois P. Heinz, Aug 15 2012

Links

Crossrefs

Column k=3 of A214776.

Programs

  • Haskell
    a126596 n = a005810 n * a005408 n `div` a016777 n
-- Reinhard Zumkeller, Mar 04 2012
  • Magma
    [Binomial(4*n,n)*(2*n+1)/(3*n+1): n in [0..20]]; // Vincenzo Librandi, Nov 18 2011
  • Maple
    seq((2*n+1)*binomial(4*n,n)/(3*n+1),n=0..22); # Emeric Deutsch, Mar 27 2007
  • Mathematica
    Table[(Binomial[4n,n](2n+1))/(3n+1),{n,0,30}] (* Harvey P. Dale, Feb 06 2016 *)

Formula

a(n) = A039599(2*n,n).
a(n) = (2*n+1)*A002293(n). - Mark van Hoeij, Nov 17 2011
a(n) = A208983(2*n+1). - Reinhard Zumkeller, Mar 04 2012
a(n) = A005810(n) * A005408(n) / A016777(n). - Reinhard Zumkeller, Mar 04 2012
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(2*n+1). - Ilya Gutkovskiy, Nov 01 2017
Recurrence: 3*n*(3*n-1)*(3*n+1)*a(n) = 8*(2*n+1)*(4*n-3)*(4*n-1)*a(n-1). - Vaclav Kotesovec, Feb 03 2018
a(n) ~ 2^(8*n+3/2) / (3^(3*n+3/2) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

Extensions

More terms from Emeric Deutsch, Mar 27 2007

A208101 Triangle read by rows: T(n,0) = 1; for n > 0: T(n,1) = n, for n>1: T(n,n) = T(n-1,n-2); T(n,k) = T(n-2,k-1) + T(n-1,k) for k: 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 1, 4, 3, 5, 2, 1, 5, 4, 9, 5, 5, 1, 6, 5, 14, 9, 14, 5, 1, 7, 6, 20, 14, 28, 14, 14, 1, 8, 7, 27, 20, 48, 28, 42, 14, 1, 9, 8, 35, 27, 75, 48, 90, 42, 42, 1, 10, 9, 44, 35, 110, 75, 165, 90, 132, 42, 1, 11, 10, 54, 44, 154, 110
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 04 2012

Keywords

Comments

Another variant of Pascal's triangle, cf. A007318.

Examples

			The triangle begins:
0:                    1
1:                  1   1
2:                1   2   1
3:              1   3   2   2
4:            1   4   3   5   2
5:          1   5   4   9   5   5
6:        1   6   5  14   9  14   5
7:      1   7   6  20  14  28  14  14
8:    1   8   7  27  20  48  28  42  14
9:  1   9   8  35  27  75  48  90  42  42

Links

Crossrefs

Cf. A208976 (row sums), A101461 (row max), A208983 (central), A208355 (right edge), A074909.

Programs

  • Haskell
    a208101 n k = a208101_tabl !! n !! k
a208101_row n = a208101_tabl !! n
a208101_tabl =  iterate
   (\row -> zipWith (+) ([0,1] ++ init row) (row ++ [0])) [1]
  • Mathematica
    T[, 0] = 1; T[n, 1] := n; T[n_, n_] := T[n-1, n-2]; T[n_, k_] /; 1Jean-François Alcover, Feb 03 2018 *)
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.