

A124428


Triangle, read by rows: T(n,k) = C([n/2],k)*C([(n+1)/2],k).


10



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
From Emeric Deutsch, Jan 18 2013: (Start)
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)


LINKS

Table of n, a(n) for n=0..71.


FORMULA

A056953(n) = Sum_{k=0..[n/2]} k!*T(n,k). A026003(n) = Sum_{k=0..[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; ...


PROG

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


CROSSREFS

Cf. A001405 (row sums), A056953, A026003, A124429 (antidiagonal sums), A124430 (eigenvector), A191521.
Columns = A002378, A006011, A006542, etc.
Sequence in context: A131034 A130313 A247073 * A191310 A124845 A191392
Adjacent sequences: A124425 A124426 A124427 * A124429 A124430 A124431


KEYWORD

nonn,tabf


AUTHOR

Paul D. Hanna, Oct 31 2006


STATUS

approved



