OFFSET
0,4
LINKS
Alois P. Heinz, Rows n = 0..500, flattened
EXAMPLE
T(6,0) = 4: [6], [2,4], [1,5], [1,2,3].
T(6,1) = 6: [5,1], [4,2], [3,1,2], [1,3,2], [2,1,3], [2,3,1].
T(6,2) = 1: [3,2,1].
T(7,0) = 5: [7], [3,4], [2,5], [1,6], [1,2,4].
T(7,1) = 7: [6,1], [4,3], [5,2], [2,1,4], [1,4,2], [2,4,1], [4,1,2].
T(7,2) = 1: [4,2,1].
Triangle T(n,k) begins:
00: 1;
01: 1;
02: 1;
03: 2, 1;
04: 2, 1;
05: 3, 2;
06: 4, 6, 1;
07: 5, 7, 1;
08: 6, 11, 2;
09: 8, 16, 3;
10: 10, 31, 15, 1;
11: 12, 36, 16, 1;
12: 15, 55, 29, 2;
13: 18, 71, 41, 3;
14: 22, 101, 65, 5;
15: 27, 147, 144, 32, 1;
MAPLE
g:= proc(u, o) option remember; `if`(u+o=0, 1, expand(
add(g(u+j-1, o-j) , j=1..o)+
add(g(u-j, o+j-1)*x, j=1..u)))
end:
b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
`if`(n>m, 0, `if`(n=m, x^i,
expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
end:
T:= n-> (p-> (q-> seq(coeff(q, x, i), i=0..degree(q)))(add(
coeff(p, x, k)*g(0, k), k=0..degree(p))))(b(n$2)):
seq(T(n), n=0..20);
MATHEMATICA
g[u_, o_] := g[u, o] = If[u+o == 0, 1, Expand[Sum[g[u+j-1, o-j], {j, 1, o}] + Sum[g[u-j, o+j-1]*x, {j, 1, u}]]]; b[n_, i_] := b[n, i] = Module[{m}, m = i*(i+1)/2; If[n>m, 0, If[n == m, x^i, Expand[b[n, i-1] + If[i>n, 0, x*b[n-i, i-1]]]]]]; T[n_] := Function [p, Function[q, Table[Coefficient[q, x, i], {i, 0, Exponent[q, x]}]][Sum[Coefficient[p, x, k]*g[0, k], {k, 0, Exponent[p, x]}]]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Apr 28 2014, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Alois P. Heinz, Apr 27 2014
STATUS
approved