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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. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
FORMULA
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
EXAMPLE
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;
...
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, t!, 0),
`if`(i<1 or n<i*(i+1)/2, 0, add(b(n-i*j, i-1, t+j)/j!, j=1..n/i)))
end:
T:= (n, k)-> b(n, k, 0):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..18);
CROSSREFS
Columns k=0-3 give: A000007, A057427, A245738, A372702.
Row sums give A107429.
Sequence in context: A029297 A289871 A257563 * A219311 A363393 A022880
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, May 25 2024
STATUS
approved

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Last modified July 24 22:37 EDT 2024. Contains 374585 sequences. (Running on oeis4.)