OFFSET
0,2
COMMENTS
3-compositions are integer compositions where up to 2 0's are allowed between successive positive parts. T(n,k) is the number of 3-compositions of n+1 having k 0's.
First column counts standard compositions.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..22800 (rows 0 <= n <= 150, flattened)
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
FORMULA
EXAMPLE
3-compositions of 2 are 2 and 1+1 with no 0's, 1+0+1 with one 0, and 1+0+0+1 with two 0's.
Triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12
0: 1
1: 2 1 1
2: 4 4 5 2 1
3: 8 12 18 13 9 3 1
4: 16 32 56 56 49 28 14 4 1
5: 32 80 160 200 210 161 105 50 20 5 1
6: 64 192 432 640 780 732 581 366 195 80 27 6 1
MATHEMATICA
Table[CoefficientList[(2 + x + x^2)^n, x], {n, 0, 8}]
PROG
(PARI) row(n) = Vecrev((x^2 + x + 2)^n); \\ Michel Marcus, Aug 14 2020
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Brian Hopkins, Aug 10 2020
STATUS
approved