

A124428


Triangle, read by rows: T(n,k) = binomial(floor(n/2),k)*binomial(floor((n+1)/2),k).


11



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, 1, 25, 100, 100, 25, 1, 1, 30, 150, 200, 75, 6, 1, 36, 225, 400, 225, 36, 1, 1, 42, 315, 700, 525, 126, 7, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 56, 588, 1960, 2450, 1176, 196, 8
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OFFSET

0,6


COMMENTS

Row sums form A001405, the central binomial coefficients: C(n,floor(n/2)). The eigenvector of this triangle is A124430.
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
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.
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)
T(n,k) is the number of noncrossing partitions of [n] having nk 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


LINKS



FORMULA

A056953(n) = Sum_{k=0..floor(n/2)} k!*T(n,k).
A026003(n) = Sum_{k=0..floor(n/2)} 2^k*T(n,k).


EXAMPLE

Triangle begins:
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;
1, 25, 100, 100, 25, 1;
1, 30, 150, 200, 75, 6;
1, 36, 225, 400, 225, 36, 1; ...


MATHEMATICA

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 *)


PROG

(PARI) T(n, k)=binomial(n\2, k)*binomial((n+1)\2, k)
(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
(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


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



