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A111301 Triangle read by rows: T(n,k) is the number of Dyck n-paths containing k even-length descents to ground level.

%I A111301 #11 Jul 28 2017 22:03:12
%S A111301 1,1,1,1,2,3,5,8,1,14,23,5,42,70,19,1,132,222,68,7,429,726,240,34,1,
%T A111301 1430,2431,847,145,9,4862,8294,3003,583,53,1,16796,28730,10712,2275,
%U A111301 262,11,58786,100776,38454,8736,1183,76,1,208012,357238,138890,33252,5068
%N A111301 Triangle read by rows: T(n,k) is the number of Dyck n-paths containing k even-length descents to ground level.
%C A111301 Column k is the sum of columns 2k and 2k+1 of A106566.
%H A111301 G. C. Greubel, <a href="/A111301/b111301.txt">Table of n, a(n) for the first 76 rows, flattened</a>
%H A111301 David Callan, <a href="https://arxiv.org/abs/math/0511010">The 136th manifestation of C_n </a>, arXiv:math/0511010 [math.CO], 2005.
%F A111301 See Mathematica line.
%F A111301 From _Emeric Deutsch_, Oct 05 2008: (Start)
%F A111301 G.f.=G(s,z)=1/[1-z(1+szC)/(1-z^2*C^2)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%F A111301 The trivariate g.f. H(t,s,z), where t (s) marks odd-length (even-length) descents to ground level and z marks semilength, is H=1/[1-z(t+szC)/(1-z^2*C^2)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. (End)
%e A111301 Table begins
%e A111301 k: ..0....1....2....3....
%e A111301 n
%e A111301 0 |..1
%e A111301 1 |..1
%e A111301 2 |..1....1
%e A111301 3 |..2....3
%e A111301 4 |..5....8....1
%e A111301 5 |.14...23....5
%e A111301 6 |.42...70...19....1
%e A111301 7 |132..222...68....7
%e A111301 a(3,1)=3 because the Dyck 3-paths containing one even-length descent to ground level are UUDUDD, UDUUDD, UUDDUD.
%t A111301 TableForm[Table[k/(n-k)Binomial[2n-2k, n]+(2k+1)/(2n-2k-1)Binomial[2n-2k-1, n], {n, 10}, {k, 0, n/2}]]
%t A111301 Join[{1}, Table[k/(n - k) Binomial[2 n - 2 k, n] + (2 k + 1)/(2 n - 2 k - 1) Binomial[2 n - 2 k - 1, n], {n, 25}, {k, 0, n/2}] // Flatten] (* _G. C. Greubel_, Jul 28 2017 *)
%Y A111301 Row sums are the Catalan numbers A000108.
%Y A111301 A143949 considers odd-length descents to the ground level. - _Emeric Deutsch_, Oct 05 2008
%K A111301 nonn,tabf
%O A111301 0,5
%A A111301 _David Callan_, Nov 02 2005