1,2

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

Alois P. Heinz, Rows n = 1..150, flattened

T(n,0) = A214257(n,0), T(n,k) = A214257(n,k)-A214257(n,k-1) for k>0.

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;

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);

# Alternative:

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

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 *)

Columns k=0-10 give: A000005, A214259, A323119, A323120, A323121, A323122, A323123, A323124, A323125, A323126, A323127.

Row sums give: A011782.

T(2n,n) gives A323111.

Cf. A214246, A214247, A214248, A214249, A214257, A214268, A214269.

Sequence in context: A208955 A121363 A346274 * A369279 A138021 A166065

Adjacent sequences: A214255 A214256 A214257 * A214259 A214260 A214261

nonn,tabl

Alois P. Heinz, Jul 08 2012

approved