A373305 -id:A373305 - cp's OEIS frontend

A373118 Number T(n,k) of compositions of n such that the set of parts is [k]; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 7, 0, 1, 11, 6, 0, 1, 20, 12, 0, 1, 32, 32, 0, 1, 54, 72, 0, 1, 87, 152, 24, 0, 1, 143, 311, 60, 0, 1, 231, 625, 180, 0, 1, 376, 1225, 450, 0, 1, 608, 2378, 1116, 0, 1, 986, 4566, 2544, 120, 0, 1, 1595, 8700, 5752, 360

Offset: 0

Alois P. Heinz, May 25 2024

			T(6,2) = 11: 1122, 1212, 1221, 2112, 2121, 2211, 11112, 11121, 11211, 12111, 21111.
T(7,3) = 12: 1123, 1132, 1213, 1231, 1312, 1321, 2113, 2131, 2311, 3112, 3121, 3211.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1,   2;
  0, 1,   3;
  0, 1,   7;
  0, 1,  11,    6;
  0, 1,  20,   12;
  0, 1,  32,   32;
  0, 1,  54,   72;
  0, 1,  87,  152,   24;
  0, 1, 143,  311,   60;
  0, 1, 231,  625,  180;
  0, 1, 376, 1225,  450;
  0, 1, 608, 2378, 1116;
  0, 1, 986, 4566, 2544, 120;
  ...

Alois P. Heinz, Rows n = 0..750, flattened

Columns k=0-3 give: A000007, A057427, A245738, A372702.
Row sums give A107429.
Cf. A000124, A000142, A000217, A000290, A001710, A003056, A008289, A332721, A371417, A373305, A373306.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, t!, 0),
     `if`(i<1 or n b(n, k, 0):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..18);

T(A000217(n),n) = n! = A000142(n).
T(A000124(n),n) = A001710(n+1) for n>=1.
T(A000290(n),n) = T(n^2,n) = A332721(n).
G.f. for column k: C({1..k},x) where C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/ (1 - Sum_{i in {s}} (x^i)) with C({},x) = 1. - John Tyler Rascoe, May 25 2024

A373306 Sum over all complete compositions of n of the element multiset size.

Original entry on oeis.org

0, 1, 2, 7, 13, 30, 73, 157, 345, 743, 1650, 3517, 7593, 16120, 34294, 72683, 153475, 323293, 679231, 1423721, 2977692, 6218395, 12959249, 26970243, 56037071, 116280086, 240953162, 498719275, 1031029386, 2129266321, 4392871427, 9054428894, 18645998093

Offset: 0

Alois P. Heinz, May 31 2024

nonn

A complete composition of n has element set [k] with k<=n (without gaps).

			a(1) = 1: 1.
a(2) = 2: 11.
a(3) = 7 = 2 + 2 + 3: 12, 21, 111.
a(4) = 13 = 3 + 3 + 3 + 4: 112, 121, 211, 1111.
a(5) = 30 = 3*3 + 4*4 + 5: 122, 212, 221, 1112, 1121, 1211, 2111, 11111.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A107429, A373118, A373305.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, [t!, 0], 0),
     `if`(i<1 or n p+[0, p[1]]*j)(
        b(n-i*j, i-1, t+j)/j!), j=1..n/i)))
    end:
a:= n-> add(b(n, k, 0)[2], k=0..floor((sqrt(1+8*n)-1)/2)):
seq(a(n), n=0..32);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[i == 0, {t!, 0}, {0, 0}], If[i < 1 || n < i*(i + 1)/2, {0, 0}, Sum[Function[p, p + {0, p[[1]]}*j][b[n - i*j, i - 1, t + j]/j!], {j, 1, n/i}]]];
a[n_] := Sum[b[n, k, 0][[2]], {k, 0, Floor[(Sqrt[1 + 8*n] - 1)/2]}];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 08 2024, after Alois P. Heinz *)

G.f.: Sum_{k>0} d/dy C({1..k},x,y)|y = 1 where C({s},x,y) = Sum_{i in {s}} (C({s}-{i},x,y)*y*x^i)/(1 - Sum_{i in {s}} (y*x^i)) with C({},x,y) = 1. - John Tyler Rascoe, Jun 18 2024

cp's OEIS Frontend

A373118 Number T(n,k) of compositions of n such that the set of parts is [k]; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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