A114687 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).
1, 1, 2, 1, 6, 4, 1, 12, 24, 8, 1, 20, 80, 80, 16, 1, 30, 200, 400, 240, 32, 1, 42, 420, 1400, 1680, 672, 64, 1, 56, 784, 3920, 7840, 6272, 1792, 128, 1, 72, 1344, 9408, 28224, 37632, 21504, 4608, 256, 1, 90, 2160, 20160, 84672, 169344, 161280, 69120, 11520
Offset: 1
Examples
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. Triangle begins: 1; 1, 2; 1, 6, 4; 1, 12, 24, 8; 1, 20, 80, 80, 16. 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
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).
- D. Callan, Polygon Dissections and Marked Dyck Paths
- G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), p. 23-24.
- G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
Programs
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Maple
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;
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Mathematica
Table[2^k*Binomial[n, k] Binomial[n, k + 1]/n, {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Nov 05 2017 *) -
PARI
t(r, m) = 2^m*binomial(r, m)*binomial(r, m+1)/r; tabl(nn) = {for (n=1, nn, for (k=0, n-1, print1(t(n,k), ", ");); print(););} \\ Michel Marcus, Nov 22 2014
Formula
T(n, k) = 2^k * binomial(n, k) * binomial(n, k+1)/n.
G.f.: G=G(t, z) satisfies G = z*(1+G)*(1+2*t*G).
Comments