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%I #37 Nov 03 2022 08:20:13
%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) = binomial(floor(n/2),k)*binomial(floor((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 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)
%C T(n,k) is the number of noncrossing partitions of [n] having n-k blocks, such that the nontrivial blocks are of type {a,b}, with a < = n/2 and b > n/2. Such partitions have k nontrivial blocks, uniquely determined by the choice of k first elements among floor(n/2) elements, and the choice of k second elements among floor((n+1)/2) elements. Indeed, by planarity, any two blocs {a,b} and {c,d} satisfy a < c iff b > d. - _Francesca Aicardi_ Nov 03 2022
%H G. C. Greubel, <a href="/A124428/b124428.txt">Rows n = 0..100 of triangle, flattened</a>
%H Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, <a href="https://arxiv.org/abs/2010.06270">Pattern statistics in faro words and permutations</a>, arXiv:2010.06270 [math.CO], 2020. See paragraph 2.1.
%F A056953(n) = Sum_{k=0..floor(n/2)} k!*T(n,k).
%F A026003(n) = Sum_{k=0..floor(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; ...
%t Table[Binomial[Floor[n/2], k]*Binomial[Floor[(n+1)/2], k], {n, 0, 15}, {k, 0, Floor[n/2]}]//Flatten (* _G. C. Greubel_, Feb 24 2019 *)
%o (PARI) T(n,k)=binomial(n\2,k)*binomial((n+1)\2,k)
%o (Magma) [[Binomial(Floor(n/2), k)*Binomial(Floor((n+1)/2),k): k in [0..Floor(n/2)]]: n in [0..15]]; // _G. C. Greubel_, Feb 24 2019
%o (Sage) [[binomial(floor(n/2),k)*binomial(floor((n+1)/2),k) for k in (0..floor(n/2))] for n in (0..15)] # _G. C. Greubel_, Feb 24 2019
%Y Cf. A001405 (row sums), A056953, A026003, A124429 (antidiagonal sums), A124430 (eigenvector), A191521.
%Y Columns = A002378, A006011, A006542, etc.
%K nonn,tabf
%O 0,6
%A _Paul D. Hanna_, Oct 31 2006
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