A373118 - OEIS

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

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OFFSET

0,8

LINKS

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

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:

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.

Cf. A000124, A000142, A000217, A000290, A001710, A003056, A008289, A332721, A371417, A373305, A373306.

Sequence in context: A029297 A289871 A257563 * A219311 A363393 A022880

Adjacent sequences: A373115 A373116 A373117 * A373119 A373120 A373121

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, May 25 2024

STATUS

approved