OFFSET
1,2
COMMENTS
For fixed k > 0, limit_{n->infinity} T(n,k)^(1/n) = d, where d > 1 is the real root of the equation d^(k+2) - 2*d^(k+1) + 1 = 0. - Vaclav Kotesovec, Jan 07 2019
LINKS
Alois P. Heinz, Rows n = 1..150, flattened
EXAMPLE
T(4,0) = 3: [4], [2,2], [1,1,1,1].
T(5,1) = 9: [3,2], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,2,2], [1,2,1,1], [1,1,2,1], [1,1,1,2].
T(5,2) = 3: [3,1,1], [1,3,1], [1,1,3].
T(5,3) = 2: [4,1], [1,4].
T(6,2) = 12: [4,2], [3,2,1], [3,1,2], [3,1,1,1], [2,4], [2,3,1], [2,1,3], [1,3,2], [1,3,1,1], [1,2,3], [1,1,3,1], [1,1,1,3].
Triangle T(n,k) begins:
1;
2, 0;
2, 2, 0;
3, 3, 2, 0;
2, 9, 3, 2, 0;
4, 11, 12, 3, 2, 0;
2, 25, 20, 12, 3, 2, 0;
4, 35, 49, 23, 12, 3, 2, 0;
MAPLE
b:= proc(n, k, s, t) option remember;
`if`(n<0, 0, `if`(n=0, 1, add(b(n-j, k,
min(s, j), max(t, j)), j=max(1, t-k+1)..s+k-1)))
end:
A:= proc(n, k) option remember;
`if`(n=0, 1, add(b(n-j, k+1, j, j), j=1..n))
end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..15);
# second Maple program:
b:= proc(n, s, t) option remember; `if`(n=0, x^(t-s),
add(b(n-j, min(s, j), max(t, j)), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n$2, 0)):
seq(T(n), n=1..15); # Alois P. Heinz, Jan 05 2019
MATHEMATICA
b[n_, k_, s_, t_] := b[n, k, s, t] = If[n < 0, 0, If[n == 0, 1, Sum [b[n-j, k, Min[s, j], Max[t, j]], {j, Max[1, t-k+1], s+k-1}]]]; A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[b[n-j, k+1, j, j], {j, 1, n}]]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 08 2012
STATUS
approved