A136621 Transpose T(n,k) of Parker's partition triangle A047812 (n >= 1 and 0 <= k <= n-1).

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 11, 20, 9, 1, 1, 18, 51, 48, 13, 1, 1, 26, 112, 169, 100, 20, 1, 1, 38, 221, 486, 461, 194, 28, 1, 1, 52, 411, 1210, 1667, 1128, 352, 40, 1, 1, 73, 720, 2761, 5095, 4959, 2517, 615, 54, 1, 1, 97, 1221, 5850, 13894, 18084, 13241, 5288, 1034, 75, 1

Offset: 1

Author

Alford Arnold, Jan 26 2008

Keywords

• nonn
• tabl

Comments

Parker's triangle is closely associated with q-binomial coefficients and Gaussian polynomials; cf. A063746. For example, row 4 of A063746 is 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, the coefficients of [8, 4], while the entries in row 4 of A047812 are the coefficients of q^(k*(4+1)) = q^(5*k) in [8, 4] where k runs from 0 to n-1 = 3. Likewise, by symmetry, "1 7 5 1" is embedded also because they are the coefficients of q^(5*(3-k)), where k runs from 0 to n-1 = 3. [Edited by Petros Hadjicostas, May 30 2020]

Examples

			Row four of A047812 is 1 5 7 1, so row four of the present entry is 1 7 5 1.
From _Petros Hadjicostas_, May 30 2020: (Start)
Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
  1;
  1,  1;
  1,  3,   1;
  1,  7,   5,    1;
  1, 11,  20,    9,    1;
  1, 18,  51,   48,   13,    1;
  1, 26, 112,  169,  100,   20,   1;
  1, 38, 221,  486,  461,  194,  28,  1;
  1, 52, 411, 1210, 1667, 1128, 352, 40, 1;
  ... (End)
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.
• Wikipedia, E. T. Parker.

Crossrefs

Cf. A000108 (Catalan row sums), A047812, A063746.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
       b((n-k-1)*(n+1), n$2):
seq(seq(T(n, k), k=0..n-1), n=1..12);  # Alois P. Heinz, May 30 2020

Mathematica

T[n_, k_]:= SeriesCoefficient[QBinomial[2*n, n, q], {q, 0, k*(n+1)}];
Table[T[n, n-k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, May 31 2020 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
T[n_, k_] := b[(n-k-1)(n+1), n, n];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

PARI

T(n, k) = #partitions(k*(n+1), n, n);
for (n=1, 10, for (k=0, n-1, print1(T(n, n-1-k), ", "); ); print(); ); \\ Petros Hadjicostas, May 30 2020
/* Second program, courtesy of G. C. Greubel */
T(n,k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
vector(12, n, vector(n, k, T(n,n-k))) \\ Petros Hadjicostas, May 31 2020

Sage

def T(n,k):
    P. = PowerSeriesRing(ZZ, k*(n+1)+1)
    return P( q_binomial(2*n, n, x) ).list()[k*(n+1)]
[[ T(n,n-k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 31 2020

Extensions

Name edited by Petros Hadjicostas, May 30 2020