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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).
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%I #20 Jul 06 2023 01:58:02

%S 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 245,63,11,1,7752,7752,3876,1428,408,88,13,1,43263,43263,21945,8379,

%U 2565,627,117,15,1,246675,246675,126500,49588,15939,4235,910,150,17,1

%N 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 From _Peter Bala_, Aug 07 2014: (Start)

%C 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 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 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 T(n,k) = C(3n-k,n-k)*(2k+1)/(2n+1) for 0<=k<=n.

%F Let M = the production matrix:

%F 1, 1

%F 2, 2, 1

%F 3, 3, 2, 1

%F 4, 4, 3, 2, 1

%F 5, 5, 4, 3, 2, 1

%F ...

%F Top row of M^(n-1) gives n-th row. - _Gary W. Adamson_, Jul 07 2011

%e Triangle begins:

%e 1;

%e 1, 1;

%e 3, 3, 1;

%e 12, 12, 5, 1;

%e 55, 55, 25, 7, 1;

%e 273, 273, 130, 42, 9, 1;

%e 1428, 1428, 700, 245, 63, 11, 1;

%e 7752, 7752, 3876, 1428, 408, 88, 13, 1; ...

%e where g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3.

%e Matrix inverse begins:

%e 1;

%e -1, 1;

%e 0, -3, 1;

%e 0, 3, -5, 1;

%e 0, -1, 10, -7, 1;

%e 0, 0, -10, 21, -9, 1;

%e 0, 0, 5, -35, 36, -11, 1;

%e 0, 0, -1, 35, -84, 55, -13, 1; ...

%e where g.f. of column k = (1-x)^(2k+1) for k>=0.

%e From _Peter Bala_, Aug 07 2014: (Start)

%e Matrix factorization as (1 + A110616)*A033184 begins

%e /1 \/ 1 \ / 1 \

%e |0 1 || 1 1 | | 1 1 |

%e |0 1 1 || 2 2 1 | = | 3 3 1 |

%e |0 3 2 1 || 5 5 3 1 | |12 12 5 1 |

%e |0 12 7 3 1 ||14 14 9 4 1 | |55 55 25 7 1 |

%e (End)

%o (PARI) {T(n,k)=binomial(3*n-k,n-k)*(2*k+1)/(2*n+1)}

%Y Cf. columns: A001764, A102893, A102594; row sums: A006013. A033184, A110616.

%K nonn,tabl

%O 1,4

%A _Paul D. Hanna_, Aug 29 2008