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A143603 Triangle, read by rows, such that the g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764 (ternary trees).

%I A143603 #20 Jul 06 2023 01:58:02
%S A143603 1,1,1,3,3,1,12,12,5,1,55,55,25,7,1,273,273,130,42,9,1,1428,1428,700,
%T A143603 245,63,11,1,7752,7752,3876,1428,408,88,13,1,43263,43263,21945,8379,
%U A143603 2565,627,117,15,1,246675,246675,126500,49588,15939,4235,910,150,17,1
%N A143603 Triangle, read by rows, such that the g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764 (ternary trees).
%C A143603 From _Peter Bala_, Aug 07 2014: (Start)
%C A143603 Riordan array (G(x), x*G(x)). Let C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + ... be the o.g.f. of the Catalan numbers A000108. Then C(x*G(x)) = G(x).
%C A143603 This leads to a factorization of this array in the group of Riordan matrices as (1, x*G(x))*(C(x), x*C(x)) = (1 + A110616)*A033184 (here, in the final product, 1 refers to the 1 X 1 identity matrix and + means direct sum - see the Example section). (End)
%H A143603 Yuxuan Zhang and Yan Zhuang, <a href="https://arxiv.org/abs/2306.15778">A subfamily of skew Dyck paths related to k-ary trees</a>, arXiv:2306.15778 [math.CO], 2023.
%F A143603 T(n,k) = C(3n-k,n-k)*(2k+1)/(2n+1) for 0<=k<=n.
%F A143603 Let M = the production matrix:
%F A143603 1, 1
%F A143603 2, 2, 1
%F A143603 3, 3, 2, 1
%F A143603 4, 4, 3, 2, 1
%F A143603 5, 5, 4, 3, 2, 1
%F A143603 ...
%F A143603 Top row of M^(n-1) gives n-th row. - _Gary W. Adamson_, Jul 07 2011
%e A143603 Triangle begins:
%e A143603 1;
%e A143603 1, 1;
%e A143603 3, 3, 1;
%e A143603 12, 12, 5, 1;
%e A143603 55, 55, 25, 7, 1;
%e A143603 273, 273, 130, 42, 9, 1;
%e A143603 1428, 1428, 700, 245, 63, 11, 1;
%e A143603 7752, 7752, 3876, 1428, 408, 88, 13, 1; ...
%e A143603 where g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3.
%e A143603 Matrix inverse begins:
%e A143603 1;
%e A143603 -1, 1;
%e A143603 0, -3, 1;
%e A143603 0, 3, -5, 1;
%e A143603 0, -1, 10, -7, 1;
%e A143603 0, 0, -10, 21, -9, 1;
%e A143603 0, 0, 5, -35, 36, -11, 1;
%e A143603 0, 0, -1, 35, -84, 55, -13, 1; ...
%e A143603 where g.f. of column k = (1-x)^(2k+1) for k>=0.
%e A143603 From _Peter Bala_, Aug 07 2014: (Start)
%e A143603 Matrix factorization as (1 + A110616)*A033184 begins
%e A143603 /1           \/ 1         \    / 1           \
%e A143603 |0  1        || 1  1       |   | 1  1        |
%e A143603 |0  1 1      || 2  2 1     | = | 3  3  1     |
%e A143603 |0  3 2 1    || 5  5 3 1   |   |12 12  5 1   |
%e A143603 |0 12 7 3 1  ||14 14 9 4 1 |   |55 55 25 7 1 |
%e A143603 (End)
%o A143603 (PARI) {T(n,k)=binomial(3*n-k,n-k)*(2*k+1)/(2*n+1)}
%Y A143603 Cf. columns: A001764, A102893, A102594; row sums: A006013. A033184, A110616.
%K A143603 nonn,tabl
%O A143603 1,4
%A A143603 _Paul D. Hanna_, Aug 29 2008