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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). 5
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS 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

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Last modified December 5 15:27 EST 2022. Contains 358588 sequences. (Running on oeis4.)