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A114656 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 peaks (1 <= k <= n).

%I A114656 #20 Apr 23 2019 17:38:42
%S A114656 1,2,1,4,6,1,8,24,12,1,16,80,80,20,1,32,240,400,200,30,1,64,672,1680,
%T A114656 1400,420,42,1,128,1792,6272,7840,3920,784,56,1,256,4608,21504,37632,
%U A114656 28224,9408,1344,72,1,512,11520,69120,161280,169344,84672,20160,2160,90
%N A114656 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 peaks (1 <= k <= n).
%C A114656 Row sums are the little Schroeder numbers (A001003). Sum_{k=1..n} k*T(n,k) = A047781(n). T(n,k) = (1/2)A114655(n,k).
%C A114656 Triangle T(n,k), 1 <= k <= n, given by [0,2,0,2,0,2,0,2,0,2,0,2,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 02 2009
%H A114656 Michael De Vlieger, <a href="/A114656/b114656.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened).
%H A114656 D. Callan, <a href="http://www.stat.wisc.edu/~callan/papers/polygon_dissections/">Polygon Dissections and Marked Dyck Paths</a>
%H A114656 Samuele Giraudo, <a href="https://arxiv.org/abs/1903.00677">Tree series and pattern avoidance in syntax trees</a>, arXiv:1903.00677 [math.CO], 2019.
%F A114656 T(n, k) = 2^(n-k)*binomial(n, k)*binomial(n, k-1)/n.
%F A114656 G.f.: G = G(t, z) satisfies G = z(2G+t)(G+1).
%F A114656 T(n,k) = A001263(n,k)*2^(n-k). - _Philippe Deléham_, Apr 11 2007
%F A114656 G.f.: 1/(1-xy/(1-2x/(1-xy/(1-2x/(1-xy/(1-2x/(1-..... (continued fraction). - _Paul Barry_, Feb 06 2009
%e A114656 T(3,2)=6 because we have (UD)Ub(UD)D, (UD)Ur(UD)D, Ub(UD)D(UD), Ur(UD)D(UD), Ub(UD)(UD)D and Ur(UD)(UD)D, where U=(1,1), D=(1,-1) and b (r) indicates a blue (red) double rise (the peaks are shown between parentheses).
%e A114656 Triangle begins:
%e A114656    1;
%e A114656    2,  1;
%e A114656    4,  6,  1;
%e A114656    8, 24, 12,  1;
%e A114656   16, 80, 80, 20,  1;
%e A114656   ....
%e A114656 Triangle T(n,k), 0 <= k <= n, given by [0,2,0,2,0,2,0,2,...] DELTA [1,0,1,0,1,0,1,0,1,0,...] begins: 1; 0,1; 0,2,1; 0,4,6,1; 0,8,24,12,1; 0,16,80,80,20,1; ... - _Philippe Deléham_, Jan 02 2009
%p A114656 T:=(n,k)->2^(n-k)*binomial(n,k)*binomial(n,k-1)/n: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%t A114656 Table[2^(n - k) Binomial[n, k] Binomial[n, k - 1]/n, {n, 10}, {k, n}] // Flatten (* _Michael De Vlieger_, Apr 23 2019 *)
%Y A114656 Cf. A001003, A047781, A114655.
%K A114656 nonn,tabl
%O A114656 1,2
%A A114656 _Emeric Deutsch_, Dec 23 2005