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 A128567 Matrix square, T(n,k), of Parker's partition triangle A047812, read by rows (n >= 1 and 0 <= k <= n-1). 3
 1, 2, 1, 5, 6, 1, 14, 31, 14, 1, 42, 133, 117, 22, 1, 132, 587, 813, 300, 36, 1, 429, 2531, 4871, 2896, 692, 52, 1, 1430, 10950, 27743, 23961, 9206, 1430, 76, 1, 4862, 47185, 151208, 175734, 96418, 24598, 2798, 104, 1, 16796, 203704, 804065, 1200301, 882471, 329426, 62885, 5236, 146, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Column 0 is the Catalan numbers (A000108). Parker's partition triangle may be defined as: A047812(n,k) = [q^(n*k+k)] in the central q-binomial coefficient [2*n,n] for n >= 1 and 0 <= k <= n-1. [Edited by Petros Hadjicostas, May 30 2020] LINKS R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy) R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289. Wikipedia, E. T. Parker. FORMULA T(n,k) = Sum_{s=k..n-1} A047812(n,s)*A047812(s+1,k) for n >= 1 and 0 <= k <= n-1. - Petros Hadjicostas, May 31 2020 EXAMPLE Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:       1;       2,      1;       5,      6,      1;      14,     31,     14,       1;      42,    133,    117,      22,      1;     132,    587,    813,     300,     36,      1;     429,   2531,   4871,    2896,    692,     52,     1;    1430,  10950,  27743,   23961,   9206,   1430,    76,    1;    4862,  47185, 151208,  175734,  96418,  24598,  2798,  104,   1;   16796, 203704, 804065, 1200301, 882471, 329426, 62885, 5236, 146, 1;   ... PROG (PARI) {T(n, k)=local(M); M=matrix(n+1, n+1, r, c, if(r

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Last modified January 24 01:20 EST 2021. Contains 340398 sequences. (Running on oeis4.)