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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy
• nice

Comments

The entries in row n are the coefficients of q^(k*(n+1)) in the q-binomial coefficient [2n, n], where k runs from 0 to n-1. - James Sellers

T(n,k) is the number of partitions of k*(n+1) into at most n parts each no bigger than n (see the links). - Petros Hadjicostas, May 30 2020

Named after the American mathematician Ernest Tilden Parker (1926-1991). - Amiram Eldar, Jun 20 2021

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) starts:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  7   1;
  1,  9, 20, 11,  1;
  1, 13, 48, 51, 18, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Richard K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
• Richard K. Guy, Parker's permutation problem involves the Catalan numbers, Preprint, 1992. (Annotated scanned copy)
• Richard K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly, Vol. 100, No. 3 (1993), pp. 287-289.
• Wikipedia, E. T. Parker.

Crossrefs

Cf. A000108 (row sums), A136621 (mirror image).

Programs

Maple

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

Mathematica

s[n_] := s[n] = Series[Product[(1-q^(2n-k)) / (1-q^(k+1)), {k, 0, n-1}], {q, 0, n^2}];
t[n_, k_] := SeriesCoefficient[s[n], k(n+1)];
Flatten[Table[t[n, k], {n, 1, 12}, {k, 0, n-1}]] (* Jean-François Alcover, Jan 27 2012 *)
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[k(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,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,k-1))); \\ Petros Hadjicostas, May 31 2020

Extensions

More terms from James Sellers

Offset corrected by Alois P. Heinz, May 30 2020