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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.

%I A336996 #18 Oct 28 2020 21:20:55
%S A336996 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,
%T A336996 160,200,210,161,105,50,20,5,1,64,192,432,640,780,732,581,366,195,80,
%U A336996 27,6,1,128,448,1120,1904,2632,2884,2674,2045,1337,721,329,119,35,7,1
%N 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.
%C A336996 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.
%C A336996 First column counts standard compositions.
%H A336996 Michael De Vlieger, <a href="/A336996/b336996.txt">Table of n, a(n) for n = 0..22800</a> (rows 0 <= n <= 150, flattened)
%H A336996 Brian Hopkins and Stéphane Ouvry, <a href="https://arxiv.org/abs/2008.04937">Combinatorics of Multicompositions</a>, arXiv:2008.04937 [math.CO], 2020.
%F A336996 T(n,k) = Sum_{m=0..n} binomial(n, m)*trinomial(m, k) using trinomial coefficients as in A027907.
%F A336996 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.
%F A336996 Row sums are powers of 4 (A000302).
%e A336996 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.
%e A336996 Triangle T(n, k) begins:
%e A336996 n\k 0   1   2   3   4   5   6   7   8   9 10 11 12
%e A336996 0:  1
%e A336996 1:  2   1   1
%e A336996 2:  4   4   5   2   1
%e A336996 3:  8  12  18  13   9   3   1
%e A336996 4: 16  32  56  56  49  28  14   4   1
%e A336996 5: 32  80 160 200 210 161 105  50  20   5  1
%e A336996 6: 64 192 432 640 780 732 581 366 195  80 27  6  1
%t A336996 Table[CoefficientList[(2 + x + x^2)^n, x], {n, 0, 8}]
%o A336996 (PARI) row(n) = Vecrev((x^2 + x + 2)^n); \\ _Michel Marcus_, Aug 14 2020
%Y A336996 Cf. A027907 for (1+x+x^2)^n, A038207 for 2-compositions.
%K A336996 nonn,tabf
%O A336996 0,2
%A A336996 _Brian Hopkins_, Aug 10 2020