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%I #17 Mar 18 2023 08:49:14
%S 1,3,1,10,4,2,35,15,9,5,126,56,36,24,14,462,210,140,100,70,42,1716,
%T 792,540,400,300,216,132,6435,3003,2079,1575,1225,945,693,429,24310,
%U 11440,8008,6160,4900,3920,3080,2288,1430,92378,43758,30888,24024,19404
%N Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)*C(2k,k)*(2k+1)/(n+k+1).
%H 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 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 Jovan Mikic, <a href="https://arxiv.org/abs/2203.12931">A Note on the Gessel Numbers</a>, arXiv:2203.12931 [math.CO], 2022.
%F T(n, k) = A000984(n)*A002457(k)/(n+k+1) = T(k, n)*(2k+1)/(2n+1).
%e Rows start:
%e 1, 3, 10, 35, 126, 462, 1716,
%e 1, 4, 15, 56, 210, 792, 3003,
%e 2, 9, 36, 140, 540, 2079, 8008,
%e 5, 24, 100, 400, 1575, 6160, 24024,
%e 14, 70, 300, 1225, 4900, 19404, 76440,
%e 42, 216, 945, 3920, 15876, 63504,252252,
%e 132, 693, 3080, 12936, 52920,213444,853776,
%e etc.
%p A078817 := proc(n,k)
%p binomial(2*n,n)*binomial(2*k,k)*(2*k+1)/(n+k+1) ;
%p end proc: # _R. J. Mathar_, Dec 06 2018
%Y Columns include A000108 (catalan), A038629, A078818 and A078819. Rows include A001700, A001791, A007946 and A078820. Diagonals include A002894 and A060150.
%Y Essentially a reflected version of A033820.
%K nonn,tabl
%O 0,2
%A _Henry Bottomley_, Dec 07 2002
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