A331332 Sparse ruler statistics: T(n,k) (0 <= k <= n) is the number of rulers with length n where the length of the first segment appears k times. Triangle read by rows.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 3, 0, 1, 0, 8, 4, 3, 0, 1, 0, 14, 9, 4, 4, 0, 1, 0, 26, 16, 12, 4, 5, 0, 1, 0, 46, 34, 21, 15, 5, 6, 0, 1, 0, 85, 64, 45, 28, 20, 6, 7, 0, 1, 0, 155, 124, 90, 64, 36, 27, 7, 8, 0, 1, 0, 286, 236, 183, 128, 90, 48, 35, 8, 9, 0, 1, 0, 528, 452, 361, 269, 185, 126, 63, 44, 9, 10, 0, 1

Offset

0,8

Comments

A sparse ruler, or simply a ruler, is a strict increasing finite sequence of nonnegative integers starting from 0 called marks. See A103294 for more definitions.

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

Sum_{k=1..n} k * T(n,k) = A175656(n-1) for n>0. - Alois P. Heinz, Feb 07 2020

Example

Triangle starts:
[ 0][1]
[ 1][0,   1]
[ 2][0,   1,   1]
[ 3][0,   3,   0,  1]
[ 4][0,   4,   3,  0,  1]
[ 5][0,   8,   4,  3,  0,  1]
[ 6][0,  14,   9,  4,  4,  0,  1]
[ 7][0,  26,  16, 12,  4,  5,  0, 1]
[ 8][0,  46,  34, 21, 15,  5,  6, 0, 1]
[ 9][0,  85,  64, 45, 28, 20,  6, 7, 0, 1]
[10][0, 155, 124, 90, 64, 36, 27, 7, 8, 0, 1]

Maple

b:= proc(n, i) option remember; `if`(n=0, x, add(expand(
     `if`(i=j, x, 1)*b(n-j, `if`(n<i+j, 0, i))), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
            `if`(n=0, 1, add(b(n-j, j), j=1..n))):
seq(T(n), n=0..12);  # Alois P. Heinz, Feb 06 2020

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, x, Sum[Expand[If[i == j, x, 1] b[n - j, If[n < i + j, 0, i]]], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ If[n == 0, 1, Sum[b[n - j, j], {j, 1, n}]]];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

Prog

(SageMath)
def A331332_row(n):
    if n == 0: return [1]
    L = [0 for k in (0..n)]
    for c in Compositions(n):
        L[list(c).count(c[0])] += 1
    return L
for n in (0..10): print(A331332_row(n))

Crossrefs

Columns k=0-1 give: A000007, A331330.

Row sums give A011782.

Row sums over even columns give A331609 (for n>0).

Row sums over odd columns give A331606 (for n>0).

T(2n,n) gives A332051.

Cf. A103294, A175656.

Sequence in context: A100257 A318315 A329861 * A300228 A100573 A049087

Adjacent sequences: A331329 A331330 A331331 * A331333 A331334 A331335

Keyword

nonn,tabl

Author

Peter Luschny, Jan 24 2020

Status

approved