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%I #20 May 31 2020 22:20:08
%S 1,2,1,5,6,1,14,31,14,1,42,133,117,22,1,132,587,813,300,36,1,429,2531,
%T 4871,2896,692,52,1,1430,10950,27743,23961,9206,1430,76,1,4862,47185,
%U 151208,175734,96418,24598,2798,104,1,16796,203704,804065,1200301,882471,329426,62885,5236,146,1
%N Matrix square, T(n,k), of Parker's partition triangle A047812, read by rows (n >= 1 and 0 <= k <= n-1).
%C 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]
%H R. K. Guy, <a href="/A007042/a007042_1.pdf">Parker's permutation problem involves the Catalan numbers</a>, preprint, 1992. (Annotated scanned copy)
%H R. K. Guy, <a href="http://www.jstor.org/stable/2324467">Parker's permutation problem involves the Catalan numbers</a>, Amer. Math. Monthly 100 (1993), 287-289.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/E._T._Parker">E. T. Parker</a>.
%F 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
%e Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
%e 1;
%e 2, 1;
%e 5, 6, 1;
%e 14, 31, 14, 1;
%e 42, 133, 117, 22, 1;
%e 132, 587, 813, 300, 36, 1;
%e 429, 2531, 4871, 2896, 692, 52, 1;
%e 1430, 10950, 27743, 23961, 9206, 1430, 76, 1;
%e 4862, 47185, 151208, 175734, 96418, 24598, 2798, 104, 1;
%e 16796, 203704, 804065, 1200301, 882471, 329426, 62885, 5236, 146, 1;
%e ...
%o (PARI) {T(n, k)=local(M);M=matrix(n+1,n+1,r,c,if(r<c,0,if(r==0,1, polcoeff(prod(j=r+1,2*r,1-q^j)/prod(j=1,r,1-q^j),(r+1)*(c-1), q)))); (M^2)[n+1,k+1]}
%o /* To display the data using the above program: */
%o vector(10, n, vector(n, k, T(n-1,k-1))) \\ _Petros Hadjicostas_, May 31 2020
%Y Cf. A000108 (column k=0), A047812, A128568 (column k=1), A128569 (column k=2), A128602 (row sums).
%K nonn,tabl
%O 1,2
%A _Paul D. Hanna_, Mar 12 2007
%E Name edited and offset changed by _Petros Hadjicostas_, May 30 2020
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