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

%I A078817 #17 Mar 18 2023 08:49:14
%S A078817 1,3,1,10,4,2,35,15,9,5,126,56,36,24,14,462,210,140,100,70,42,1716,
%T A078817 792,540,400,300,216,132,6435,3003,2079,1575,1225,945,693,429,24310,
%U A078817 11440,8008,6160,4900,3920,3080,2288,1430,92378,43758,30888,24024,19404
%N A078817 Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)*C(2k,k)*(2k+1)/(n+k+1).
%H A078817 Ira Gessel, <a href="https://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf">Super ballot numbers</a>,  J. Symbolic Computation 14 (1992), 179-194.
%H A078817 Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
%H A078817 Jovan Mikic, <a href="https://arxiv.org/abs/2203.12931">A Note on the Gessel Numbers</a>, arXiv:2203.12931 [math.CO], 2022.
%F A078817 T(n, k) = A000984(n)*A002457(k)/(n+k+1) = T(k, n)*(2k+1)/(2n+1).
%e A078817 Rows start:
%e A078817      1,     3,    10,    35,   126,   462,  1716,
%e A078817      1,     4,    15,    56,   210,   792,  3003,
%e A078817      2,     9,    36,   140,   540,  2079,  8008,
%e A078817      5,    24,   100,   400,  1575,  6160, 24024,
%e A078817     14,    70,   300,  1225,  4900, 19404, 76440,
%e A078817     42,   216,   945,  3920, 15876, 63504,252252,
%e A078817    132,   693,  3080, 12936, 52920,213444,853776,
%e A078817 etc.
%p A078817 A078817 := proc(n,k)
%p A078817     binomial(2*n,n)*binomial(2*k,k)*(2*k+1)/(n+k+1) ;
%p A078817 end proc: # _R. J. Mathar_, Dec 06 2018
%Y A078817 Columns include A000108 (catalan), A038629, A078818 and A078819. Rows include A001700, A001791, A007946 and A078820. Diagonals include A002894 and A060150.
%Y A078817 Essentially a reflected version of A033820.
%K A078817 nonn,tabl
%O A078817 0,2
%A A078817 _Henry Bottomley_, Dec 07 2002