A245738 -id:A245738 - 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

A372702 Number of compositions of n such that the set of parts is {1,2,3}.

Original entry on oeis.org

6, 12, 32, 72, 152, 311, 625, 1225, 2378, 4566, 8700, 16475, 31052, 58290, 109079, 203584, 379144, 704821, 1308268, 2425259, 4491074, 8308879, 15360082, 28376089, 52391492, 96683649, 178344205, 328854566, 606190627, 1117103729, 2058129088, 3791056189

Offset: 6

John Tyler Rascoe, May 25 2024

Alois P. Heinz, Table of n, a(n) for n = 6..3779

Cf. A048793, A107429, A245738.

Column k=3 of A373118.

b:= proc(n, t) option remember; `if`(n=0, `if`(t=7, 1, 0),
      add(b(n-j, Bits[Or](t, 2^(j-1))), j=1..min(n, 3)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=6..42);  # Alois P. Heinz, May 25 2024

C_x(s,N)={my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, C_x(setminus(s,[s[i]]),N) * x^(s[i]) )/(1-sum(i=1,#s, x^(s[i]))))); return(g)}
B_x(n) ={my(h=C_x([1,2,3],n)); Vec(h)}
B_x(40)

G.f.: C({1,2,3},x) = (x^6/(-x^3 - x^2 - x + 1)) *
 (1/((1 - x)*(-x^2 - x + 1)) +
  1/((1 - x)*(-x^3 - x + 1)) +
  1/((1 - x^2)*(-x^2 - x + 1)) +
  1/((1 - x^2)*(-x^3 - x^2 + 1)) +
  1/((1 - x^3)*(-x^3 - x + 1)) +
  1/((1 - x^3)*(-x^3 - x^2 + 1))).
Where C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)).

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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