Search: id:A051288
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%I A051288
%S A051288 1,2,4,2,8,12,16,48,6,32,160,60,64,480,360,20,128,1344,1680,280,256,
%T A051288 3584,6720,2240,70,512,9216,24192,13440,1260,1024,23040,80640,67200,
%U A051288 12600,252,2048,56320,253440,295680,92400,5544,4096,135168,760320
%N A051288 Triangle read by rows: a(n,k) = number of paths of n upsteps U and n 
               downsteps D that contain k UUDs.
%C A051288 By reading paths backward, the UUD in the name could be replaced by DDU.
%C A051288 Or, triangular array T read by rows: T(n,k)=P(2n,n,4k), where P(n,k,c)=number 
               of vectors (x(1),x(2,),...,x(n)) of k 1's and n-k 0's such that x(i)=x(n+1-i) 
               for exactly c values of i. P(n,k,n) counts palindromes.
%F A051288 a(n, k)=binom(n, 2k)2^(n-2k)binom(2k, k). G.f. (1-4x+4x^2(1-y))^(-1/2) 
               = Sum_{n>=0, k>=0} a(n, k) x^n y^k.
%e A051288 Table begins
%e A051288 \ k 0, 1, 2, ...
%e A051288 n
%e A051288 0 | 1
%e A051288 1 | 2
%e A051288 2 | 4, 2
%e A051288 3 | 8, 12,
%e A051288 4 | 16, 48, 6
%e A051288 5 | 32, 160, 60
%e A051288 6 | 64, 480, 360, 20
%e A051288 7 |128, 1344, 1680, 280
%e A051288 a(2,1)=2 because UUDD, DUUD each have one UUD.
%t A051288 Table[Binomial[n, 2k]2^(n-2k)Binomial[2k, k], {n, 0, 15}, {k, 0, n/2}]
%Y A051288 Row sums are the (even) central binomial coefficients A000984. A091894 
               gives the distribution of the parameter "number of DDUs" on Dyck 
               paths.
%Y A051288 Sequence in context: A068217 A114593 A114655 this_sequence A120434 A008303 
               A058942
%Y A051288 Adjacent sequences: A051285 A051286 A051287 this_sequence A051289 A051290 
               A051291
%K A051288 nonn,tabf
%O A051288 0,2
%A A051288 Clark Kimberling (ck6(AT)evansville.edu)
%E A051288 Additional comments from David Callan (callan(AT)stat.wisc.edu), Aug 
               28 2004

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