

A191312


Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having abscissa of the first return to the horizontal axis equal to k (assumed to be 0 if there are no such returns).


1



1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 0, 3, 2, 2, 2, 1, 0, 6, 3, 4, 2, 4, 1, 0, 10, 6, 6, 4, 4, 4, 1, 0, 20, 10, 12, 6, 8, 4, 9, 1, 0, 35, 20, 20, 12, 12, 8, 9, 9, 1, 0, 70, 35, 40, 20, 24, 12, 18, 9, 23, 1, 0, 126, 70, 70, 40, 40, 24, 27, 18, 23, 23, 1, 0, 252, 126, 140, 70, 80, 40, 54, 27, 46, 23, 65
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OFFSET

0,13


COMMENTS

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
Sum_{k>=0} k*T(n,k) = A093387(n+1).


LINKS

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


FORMULA

T(n,0)=1; T(n,1)=0;
T(n,k) = binomial(nk, floor((nk)/2))*Sum_{j=0..floor(k/2)1} c(j), where 2<=k<=n and c(j) = binomial(2*j,j)/(j+1) are the Catalan numbers.
G.f.: G(t,z) = 1/(1z)+(1sqrt(14*t^2*z^2))/((1t*z)*(12*z+sqrt(14*z^2)).
For k>=1, g.f. of column 2k is b_{k1}*z^{2k}*g and of column 2k+1 is b_{k1}*z^{2*k+1}*g, where g = 2/(12*z+sqrt(14*z^2)) and b(k) = Sum_{j=0..k1} c(j) with c(j) = binomial(2*j,j)/(j+1) = A000108(j) (the Catalan numbers).


EXAMPLE

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


MAPLE

c := proc (j) options operator, arrow: binomial(2*j, j)/(j+1) end proc: T := proc (n, k) if n < k then 0 elif k = 0 then 1 elif k = 1 then 0 else binomial(nk, floor((1/2)*n(1/2)*k))*(sum(c(j), j = 0 .. floor((1/2)*k)1)) 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
G := (1t*z+t^2*z^2*g*Ct^2*z^3*g*C)/((1z)*(1t*z)): g := 2/(12*z+sqrt(14*z^2)): C := ((1sqrt(14*t^2*z^2))*1/2)/(t^2*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


CROSSREFS

Cf. A001405, A191313.
Sequence in context: A159817 A079532 A328176 * A240159 A309447 A320312
Adjacent sequences: A191309 A191310 A191311 * A191313 A191314 A191315


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, May 30 2011


STATUS

approved



