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A047812
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Parker's partition triangle T(n,k) read by rows (n >= 1 and 0 <= k <= n-1).
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15
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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
(list;
table;
graph;
refs;
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history;
text;
internal format)
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OFFSET
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1,5
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COMMENTS
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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
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LINKS
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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.
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EXAMPLE
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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;
...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000108 (row sums), A136621 (mirror image).
Sequence in context: A099608 A247285 A047969 * A129392 A118538 A141523
Adjacent sequences: A047809 A047810 A047811 * A047813 A047814 A047815
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from James A. Sellers
Offset corrected by Alois P. Heinz, May 30 2020
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STATUS
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approved
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