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A336996
Triangle of coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (2 + x + x^2)^n.
2
1, 2, 1, 1, 4, 4, 5, 2, 1, 8, 12, 18, 13, 9, 3, 1, 16, 32, 56, 56, 49, 28, 14, 4, 1, 32, 80, 160, 200, 210, 161, 105, 50, 20, 5, 1, 64, 192, 432, 640, 780, 732, 581, 366, 195, 80, 27, 6, 1, 128, 448, 1120, 1904, 2632, 2884, 2674, 2045, 1337, 721, 329, 119, 35, 7, 1
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
T(n,k) = Sum_{m=0..n} binomial(n, m)*trinomial(m, k) using trinomial coefficients as in A027907.
Recurrence: T(0,0) = 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) + 2*T(n-1,k), with T(n,k) = 0 if k < 0 or k > 2*n.
Row sums are powers of 4 (A000302).
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
Cf. A027907 for (1+x+x^2)^n, A038207 for 2-compositions.
Sequence in context: A346031 A129704 A144460 * A222541 A177263 A357050
KEYWORD
nonn,tabf
AUTHOR
Brian Hopkins, Aug 10 2020
STATUS
approved