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

1,5

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 A. 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

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.

Example

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;
  ...

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
      <n, 0, b(n, i-1, t)+b(n-i, min(i, n-i), t-1)))
    end:
T:= (n, k)-> 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 *)

Prog

(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

Crossrefs

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

Sequence in context: A099608 A247285 A047969 * A129392 A118538 A141523

Adjacent sequences: A047809 A047810 A047811 * A047813 A047814 A047815

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers

Offset corrected by Alois P. Heinz, May 30 2020

Status

approved