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Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).
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%I #33 Jun 03 2018 03:48:26

%S 1,1,2,1,6,4,1,12,24,8,1,20,80,80,16,1,30,200,400,240,32,1,42,420,

%T 1400,1680,672,64,1,56,784,3920,7840,6272,1792,128,1,72,1344,9408,

%U 28224,37632,21504,4608,256,1,90,2160,20160,84672,169344,161280,69120,11520

%N Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).

%C Row sums are the little Schroeder numbers (A001003). Sum(k*T(n,k),k=0..n-1) = 2*A050152(n-1). Mirror image of A114656.

%C Triangle T(n,k) given (essentially) by [1,0,1,0,1,0,1,0,1,0,1,0,...] DELTA [0,2,0,2,0,2,0,2,0,2,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 02 2009

%C T(r, m) is the number distinct extremities of the [0,r]-covering hierarchies with segments terminating at r (see Kreweras work). - _Michel Marcus_, Nov 22 2014

%H Michael De Vlieger, <a href="/A114687/b114687.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150).

%H D. Callan, <a href="http://www.stat.wisc.edu/~callan/papers/polygon_dissections/">Polygon Dissections and Marked Dyck Paths</a>

%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), p. 23-24.

%H G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

%F T(n, k) = 2^k * binomial(n, k) * binomial(n, k+1)/n.

%F G.f.: G=G(t, z) satisfies G = z*(1+G)*(1+2*t*G).

%e T(3,2)=4 because we have UbUbUDDD, UbUrUDDD, UrUbUDDD and UrUrUDDD, where U=(1,1), D=(1,-1) and b (r) indicates a blue (red) double rise.

%e Triangle begins:

%e 1;

%e 1, 2;

%e 1, 6, 4;

%e 1, 12, 24, 8;

%e 1, 20, 80, 80, 16.

%e Triangle [1,0,1,0,1,0,1,0,...] DELTA [0,2,0,2,0,2,0,2,0,...]:= T(n,k), 0 <= k <= n, begins: 1; 1,0; 1,2,0; 1,6,4,0; 1,12,24,8,0; 1,20,80,80,16,0; ... - _Philippe Deléham_, Jan 02 2009

%p T:=(n,k)->2^k*binomial(n,k)*binomial(n,k+1)/n: for n from 1 to 11 do seq(T(n,k),k=0..n-1) od;

%t Table[2^k*Binomial[n, k] Binomial[n, k + 1]/n, {n, 10}, {k, 0, n - 1}] // Flatten (* _Michael De Vlieger_, Nov 05 2017 *)

%o (PARI) t(r, m) = 2^m*binomial(r, m)*binomial(r, m+1)/r;

%o tabl(nn) = {for (n=1, nn, for (k=0, n-1, print1(t(n,k), ", ");); print(););} \\ _Michel Marcus_, Nov 22 2014

%Y Cf. A001003, A050152, A114656.

%K nonn,tabl

%O 1,3

%A _Emeric Deutsch_, Dec 23 2005