

A191530


Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of semilength n (i.e., Motzkin paths of length n with no (1,0)steps at positive heights) for which the sum of the lengths of the initial and final horizontal segments is k (0<=k<=n).


1



1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 1, 4, 0, 4, 0, 1, 6, 2, 6, 0, 5, 0, 1, 5, 12, 3, 8, 0, 6, 0, 1, 20, 10, 18, 4, 10, 0, 7, 0, 1, 21, 40, 15, 24, 5, 12, 0, 8, 0, 1, 70, 42, 60, 20, 30, 6, 14, 0, 9, 0, 1, 84, 140, 63, 80, 25, 36, 7, 16, 0, 10, 0, 1, 252, 168, 210, 84, 100, 30, 42, 8, 18, 0, 11, 0, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,8


COMMENTS

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
T(n,0) = A191529(n).
Sum_{k>=0} k*T(n,k) = A191531(n).


LINKS

Table of n, a(n) for n=0..90.


FORMULA

T(2n,0) = binomial(2n2,n1) (n>=1); T(2n+1,0) = binomial(2n1,n2) (n>=1).
T(n,k) = (k+1)T(nk,0).
G.f.: G(t,z) = (2  3z  tz + 2tz^2 + (1t)z*sqrt(14z^2))/((1  2z + sqrt(14z^2))(1tz)^2).


EXAMPLE

T(5,3)=4 because we have HHHUD, HHUDH, HUDHH, and UDHHH, where U=(1,1), H=(1,0), and D=(1,1).
Triangle starts:
1;
0, 1;
1, 0, 1;
0, 2, 0, 1;
2, 0, 3, 0, 1;
1, 4, 0, 4, 0, 1;
6, 2, 6, 0, 5, 0, 1;


MAPLE

q := sqrt(14*z^2): G := (23*zt*z+2*t*z^2+(1t)*z*q)/((12*z+q)*(1t*z)^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
T := proc (n, k) if n < k then 0 elif k = n then 1 elif k = 0 and n = 1 then 0 elif k = 0 and `mod`(n, 2) = 0 then binomial(n2, (1/2)*n1) elif k = 0 and `mod`(n, 2) = 1 then binomial(n2, (1/2)*n5/2) else (1+k)*T(nk, 0) end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form


CROSSREFS

Cf. A001405, A191529, A191531.
Sequence in context: A053121 A113408 A242653 * A173863 A022337 A025687
Adjacent sequences: A191527 A191528 A191529 * A191531 A191532 A191533


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Jun 07 2011


STATUS

approved



