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%I
%S 1,1,1,1,1,2,1,4,1,1,6,3,1,9,9,1,1,12,18,4,1,16,36,16,1,1,20,60,40,5,
%T 1,25,100,100,25,1,1,30,150,200,75,6,1,36,225,400,225,36,1,1,42,315,
%U 700,525,126,7,1,49,441,1225,1225,441,49,1,1,56,588,1960,2450,1176,196,8
%N Triangle, read by rows: T(n,k) = C([n/2],k)*C([(n+1)/2],k).
%C Row sums form A001405, the central binomial coefficients: C(n,floor(n/2)). The eigenvector of this triangle is A124430.
%C T(n,k) is the number of dispersed Dyck paths of length n (i.e. Motzkin paths of length n with no (1,0) steps at positive heights) having k peaks. Example: T(5,2)=3 because, denoting U=(1,1), D=(1,-1), H=1,0), we have HUDUD, UDHUD, and UDUDH. [Emeric Deutsch, Jun 01 2011]
%C Contribution from Emeric Deutsch, Jan 18 2013. (Start)
%C T(n,k) is the number of Dyck prefixes of length n having k peaks. Example: T(5,2)=3 because we have (UD)(UD)U, (UD)U(UD), and U(UD)(UD); the peaks are shown between parentheses.
%C T(n,k) is the number of Dyck prefixes of length n having k ascents and descents of length >=2. Example: T(5,2)=3 because we have (UU)(DD)U, (UU)D(UU), and (UUU)(DD); the ascents and descents of length >=2 are shown between parentheses. (end)
%F A056953(n) = Sum_{k=0..[n/2]} k!*T(n,k). A026003(n) = Sum_{k=0..[n/2]} 2^k*T(n,k).
%e Triangle begins:
%e 1;
%e 1;
%e 1, 1;
%e 1, 2;
%e 1, 4, 1;
%e 1, 6, 3;
%e 1, 9, 9, 1;
%e 1, 12, 18, 4;
%e 1, 16, 36, 16, 1;
%e 1, 20, 60, 40, 5;
%e 1, 25, 100, 100, 25, 1;
%e 1, 30, 150, 200, 75, 6;
%e 1, 36, 225, 400, 225, 36, 1; ...
%o (PARI) T(n,k)=binomial(n\2,k)*binomial((n+1)\2,k)
%Y Cf. A001405 (row sums), A056953, A026003, A124429 (antidiagonal sums), A124430 (eigenvector).
%K nonn,tabf
%O 0,6
%A _Paul D. Hanna_, Oct 31 2006
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