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%I #36 Dec 28 2019 10:18:25
%S 1,1,1,3,2,1,12,7,3,1,55,30,12,4,1,273,143,55,18,5,1,1428,728,273,88,
%T 25,6,1,7752,3876,1428,455,130,33,7,1,43263,21318,7752,2448,700,182,
%U 42,8,1,246675,120175,43263,13566,3876,1020,245,52,9,1
%N A convolution triangle of numbers based on A001764.
%C Reflected version of A069269. - _Vladeta Jovovic_, Sep 27 2006
%C With offset 1 for n and k, T(n,k) = number of Dyck paths of semilength n for which all descents are of even length (counted by A001764) with no valley vertices at height 1 and with k returns to ground level. For example, T(3,2)=2 counts U^4 D^4 U^2 D^2, U^2 D^2 U^4 D^4 where U=upstep, D=downstep and exponents denote repetition. - _David Callan_, Aug 27 2009
%C Riordan array (f(x), x*f(x)) with f(x) = (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))). - _Philippe Deléham_, Jan 27 2014
%C Antidiagonals of convolution matrix of Table 1.4, p. 397, of Hoggatt and Bicknell. - _Tom Copeland_, Dec 25 2019
%H Naiomi Cameron, J. E. McLeod, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/McLeod/mcleod3.html">Returns and Hills on Generalized Dyck Paths</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
%H V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.
%H Sheng-Liang Yang, LJ Wang, <a href="https://ajc.maths.uq.edu.au/pdf/64/ajc_v64_p420.pdf">Taylor expansions for the m-Catalan numbers</a>, Australasian Journal of Combinatorics, Volume 64(3) (2016), Pages 420-431.
%F T(n, k) = Sum_{j>=0} T(n-1, k-1+j)*A000108(j); T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n.
%F G.f.: 1/(1 - x*y*TernaryGF) = 1 + (y)x + (y+y^2)x^2 + (3y+2y^2+y^3)x^3 +... where TernaryGF = 1 + x + 3x^2 + 12x^3 + ... is the GF for A001764. - _David Callan_, Aug 27 2009
%F T(n, k) = ((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1). - _Vladimir Kruchinin_, Nov 01 2011
%e Triangle begins:
%e 1;
%e 1, 1;
%e 3, 2, 1;
%e 12, 7, 3, 1;
%e 55, 30, 12, 4, 1;
%e 273, 143, 55, 18, 5, 1;
%e 1428, 728, 273, 88, 25, 6, 1;
%e 7752, 3876, 1428, 455, 130, 33, 7, 1;
%e 43263, 21318, 7752, 2448, 700, 182, 42, 8, 1;
%e 246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1;
%e ...
%t Table[(k + 1) Binomial[3 n - 2 k, 2 n - k]/(2 n - k + 1), {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Jun 28 2017 *)
%o (Maxima) T(n,k):=((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1); // _Vladimir Kruchinin_, Nov 01 2011
%Y Successive columns: A001764, A006013, A001764, A006629, A102893, A006630, A102594, A006631; row sums: A098746; see also A092276.
%K nonn,tabl
%O 0,4
%A _Philippe Deléham_, Sep 14 2005, Jun 15 2007
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