A078817 Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)*C(2k,k)*(2k+1)/(n+k+1).

1, 3, 1, 10, 4, 2, 35, 15, 9, 5, 126, 56, 36, 24, 14, 462, 210, 140, 100, 70, 42, 1716, 792, 540, 400, 300, 216, 132, 6435, 3003, 2079, 1575, 1225, 945, 693, 429, 24310, 11440, 8008, 6160, 4900, 3920, 3080, 2288, 1430, 92378, 43758, 30888, 24024, 19404

Offset

0,2

Links

Table of n, a(n) for n=0..49.

Ira Gessel, Super ballot numbers, J. Symbolic Computation 14 (1992), 179-194.

Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.

Jovan Mikic, A Note on the Gessel Numbers, arXiv:2203.12931 [math.CO], 2022.

Formula

T(n, k) = A000984(n)*A002457(k)/(n+k+1) = T(k, n)*(2k+1)/(2n+1).

Example

Rows start:
     1,     3,    10,    35,   126,   462,  1716,
     1,     4,    15,    56,   210,   792,  3003,
     2,     9,    36,   140,   540,  2079,  8008,
     5,    24,   100,   400,  1575,  6160, 24024,
    14,    70,   300,  1225,  4900, 19404, 76440,
    42,   216,   945,  3920, 15876, 63504,252252,
   132,   693,  3080, 12936, 52920,213444,853776,
etc.

Maple

A078817 := proc(n, k)
    binomial(2*n, n)*binomial(2*k, k)*(2*k+1)/(n+k+1) ;
end proc: # R. J. Mathar, Dec 06 2018

Crossrefs

Columns include A000108 (catalan), A038629, A078818 and A078819. Rows include A001700, A001791, A007946 and A078820. Diagonals include A002894 and A060150.

Essentially a reflected version of A033820.

Sequence in context: A176992 A319375 A107870 * A316193 A091042 A111418

Adjacent sequences: A078814 A078815 A078816 * A078818 A078819 A078820

Keyword

nonn,tabl

Author

Henry Bottomley, Dec 07 2002

Status

approved