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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).

%I A128567 #20 May 31 2020 22:20:08
%S A128567 1,2,1,5,6,1,14,31,14,1,42,133,117,22,1,132,587,813,300,36,1,429,2531,
%T A128567 4871,2896,692,52,1,1430,10950,27743,23961,9206,1430,76,1,4862,47185,
%U A128567 151208,175734,96418,24598,2798,104,1,16796,203704,804065,1200301,882471,329426,62885,5236,146,1
%N A128567 Matrix square, T(n,k), of Parker's partition triangle A047812, read by rows (n >= 1 and 0 <= k <= n-1).
%C A128567 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 A128567 R. K. Guy, <a href="/A007042/a007042_1.pdf">Parker's permutation problem involves the Catalan numbers</a>, preprint, 1992. (Annotated scanned copy)
%H A128567 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 A128567 Wikipedia, <a href="https://en.wikipedia.org/wiki/E._T._Parker">E. T. Parker</a>.
%F A128567 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 A128567 Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
%e A128567       1;
%e A128567       2,      1;
%e A128567       5,      6,      1;
%e A128567      14,     31,     14,       1;
%e A128567      42,    133,    117,      22,      1;
%e A128567     132,    587,    813,     300,     36,      1;
%e A128567     429,   2531,   4871,    2896,    692,     52,     1;
%e A128567    1430,  10950,  27743,   23961,   9206,   1430,    76,    1;
%e A128567    4862,  47185, 151208,  175734,  96418,  24598,  2798,  104,   1;
%e A128567   16796, 203704, 804065, 1200301, 882471, 329426, 62885, 5236, 146, 1;
%e A128567   ...
%o A128567 (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 A128567 /* To display the data using the above program: */
%o A128567 vector(10, n, vector(n, k, T(n-1,k-1))) \\ _Petros Hadjicostas_, May 31 2020
%Y A128567 Cf. A000108 (column k=0), A047812, A128568 (column k=1), A128569 (column k=2), A128602 (row sums).
%K A128567 nonn,tabl
%O A128567 1,2
%A A128567 _Paul D. Hanna_, Mar 12 2007
%E A128567 Name edited and offset changed by _Petros Hadjicostas_, May 30 2020