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A136621 Transpose T(n,k) of Parker's partition triangle A047812 (n >= 1 and 0 <= k <= n-1).

%I A136621 #37 Nov 27 2020 23:43:13
%S A136621 1,1,1,1,3,1,1,7,5,1,1,11,20,9,1,1,18,51,48,13,1,1,26,112,169,100,20,
%T A136621 1,1,38,221,486,461,194,28,1,1,52,411,1210,1667,1128,352,40,1,1,73,
%U A136621 720,2761,5095,4959,2517,615,54,1,1,97,1221,5850,13894,18084,13241,5288,1034,75,1
%N A136621 Transpose T(n,k) of Parker's partition triangle A047812 (n >= 1 and 0 <= k <= n-1).
%C A136621 Parker's triangle is closely associated with q-binomial coefficients and Gaussian polynomials; cf. A063746. For example, row 4 of A063746 is 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, the coefficients of [8, 4], while the entries in row 4 of A047812 are the coefficients of q^(k*(4+1)) = q^(5*k) in [8, 4] where k runs from 0 to n-1 = 3. Likewise, by symmetry, "1 7 5 1" is embedded also because they are the coefficients of q^(5*(3-k)), where k runs from 0 to n-1 = 3. [Edited by _Petros Hadjicostas_, May 30 2020]
%H A136621 Alois P. Heinz, <a href="/A136621/b136621.txt">Rows n = 1..141, flattened</a>
%H A136621 R. K. Guy, <a href="http://www.jstor.org/stable/2324467">Parker's permutation problem involves the Catalan numbers</a>, Amer. Math. Monthly 100 (1993), 287-289.
%H A136621 Wikipedia, <a href="https://en.wikipedia.org/wiki/E._T._Parker">E. T. Parker</a>.
%e A136621 Row four of A047812 is 1 5 7 1, so row four of the present entry is 1 7 5 1.
%e A136621 From _Petros Hadjicostas_, May 30 2020: (Start)
%e A136621 Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
%e A136621   1;
%e A136621   1,  1;
%e A136621   1,  3,   1;
%e A136621   1,  7,   5,    1;
%e A136621   1, 11,  20,    9,    1;
%e A136621   1, 18,  51,   48,   13,    1;
%e A136621   1, 26, 112,  169,  100,   20,   1;
%e A136621   1, 38, 221,  486,  461,  194,  28,  1;
%e A136621   1, 52, 411, 1210, 1667, 1128, 352, 40, 1;
%e A136621   ... (End)
%p A136621 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
%p A136621       <n, 0, b(n, i-1, t)+b(n-i, min(i, n-i), t-1)))
%p A136621     end:
%p A136621 T:= (n, k)-> b((n-k-1)*(n+1), n$2):
%p A136621 seq(seq(T(n, k), k=0..n-1), n=1..12);  # _Alois P. Heinz_, May 30 2020
%t A136621 T[n_, k_]:= SeriesCoefficient[QBinomial[2*n, n, q], {q, 0, k*(n+1)}];
%t A136621 Table[T[n, n-k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, May 31 2020 *)
%t A136621 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 A136621 T[n_, k_] := b[(n-k-1)(n+1), n, n];
%t A136621 Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)
%o A136621 (PARI) T(n, k) = #partitions(k*(n+1), n, n);
%o A136621 for (n=1, 10, for (k=0, n-1, print1(T(n, n-1-k), ", "); ); print(); ); \\ _Petros Hadjicostas_, May 30 2020
%o A136621 /* Second program, courtesy of _G. C. Greubel_ */
%o A136621 T(n,k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
%o A136621 vector(12, n, vector(n, k, T(n,n-k))) \\ _Petros Hadjicostas_, May 31 2020
%o A136621 (Sage)
%o A136621 def T(n,k):
%o A136621     P.<x> = PowerSeriesRing(ZZ, k*(n+1)+1)
%o A136621     return P( q_binomial(2*n, n, x) ).list()[k*(n+1)]
%o A136621 [[ T(n,n-k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, May 31 2020
%Y A136621 Cf. A000108 (Catalan row sums), A047812, A063746.
%K A136621 nonn,tabl
%O A136621 1,5
%A A136621 _Alford Arnold_, Jan 26 2008
%E A136621 Name edited by _Petros Hadjicostas_, May 30 2020