A110616 - OEIS

%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