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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.
3
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
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.
Sequence in context: A318315 A373951 A329861 * A300228 A100573 A049087
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 24 2020
STATUS
approved