A128567 Matrix square, T(n,k), of Parker's partition triangle A047812, read by rows (n >= 1 and 0 <= k <= n-1).

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

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

Table of n, a(n) for n=1..55.

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<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]}
/* To display the data using the above program: */
vector(10, n, vector(n, k, T(n-1, k-1))) \\ Petros Hadjicostas, May 31 2020

Crossrefs

Cf. A000108 (column k=0), A047812, A128568 (column k=1), A128569 (column k=2), A128602 (row sums).

Sequence in context: A047887 A120986 A095801 * A345394 A217204 A179455

Adjacent sequences: A128564 A128565 A128566 * A128568 A128569 A128570

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 12 2007

Extensions

Name edited and offset changed by Petros Hadjicostas, May 30 2020

Status

approved