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
KEYWORD
AUTHOR
EXTENSIONS
More terms from James A. Sellers
Offset corrected by Alois P. Heinz, May 30 2020
STATUS
approved