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 A047812 Parker's partition triangle T(n,k) read by rows (n >= 1 and 0 <= k <= n-1). 15
 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; listen; history; text; internal format)
 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 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

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Last modified September 9 07:24 EDT 2024. Contains 375762 sequences. (Running on oeis4.)