A331609 -id:A331609 - cp's OEIS frontend

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

Peter Luschny, Jan 24 2020

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.

			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]

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

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.

b:= proc(n, i) option remember; `if`(n=0, x, add(expand(
     `if`(i=j, x, 1)*b(n-j, `if`(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

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 *)

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))

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

A331606 Number of compositions of n with the multiplicity of the first part odd.

Original entry on oeis.org

1, 1, 4, 4, 12, 18, 44, 72, 158, 288, 604, 1146, 2332, 4528, 9126, 17944, 35940, 71130, 142132, 282344, 563630, 1121936, 2239060, 4462530, 8906236, 17764160, 35458774, 70761520, 141272876, 282025466, 563159588, 1124543256, 2245918406, 4485670168, 8960061076

Offset: 1

Arnold Knopfmacher, Jan 22 2020

nonn

			For n=3, a(4)=4 as we count 4, 3+1, 1+3 and 2+1+1.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.

Cf. A011782, A331609 (similar, with even).

b:= proc(n, p, t) option remember; `if`(n=0, t,
      add(b(n-j, p, `if`(p=j, 1-t, t)), j=1..n))
    end:
a:= n-> add(b(n-j, j, 1), j=1..n):
seq(a(n), n=1..38);  # Alois P. Heinz, Jan 23 2020

gf[x_] := x/(2 (1 - 2 x)) + Sum[(1 - x) x^i/(2 (-2 x^(i + 1) + 2 x^i - 2 x + 1))  , {i, 1, 40}]; CL := CoefficientList[Series[gf[x], {x, 0, 35}], x];
Drop[CL, 1] (* Peter Luschny, Jan 23 2020 *)

G.f.: Sum_{i>=1} (1-x)*x^i/(2*(-2*x^(i+1)+2*x^i-2*x+1)) + x/(2*(1-2*x)).

a(n) = A011782(n) - A331609(n). - Alois P. Heinz, Jan 23 2020

cp's OEIS Frontend

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.

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