login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A143952 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peak plateaux (0<=k<=floor(n/2)). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep. 2

%I #7 Jun 11 2017 08:34:38

%S 1,1,1,1,1,4,1,12,1,1,33,8,1,88,42,1,1,232,183,13,1,609,717,102,1,1,

%T 1596,2622,624,19,1,4180,9134,3275,205,1,1,10945,30691,15473,1650,26,

%U 1,28656,100284,67684,11020,366,1,1,75024,320466,279106,64553,3716,34,1

%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peak plateaux (0<=k<=floor(n/2)). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.

%C Row n has 1+floor(n/2) terms.

%C Row sums are the Catalan numbers (A000108).

%C T(n,1)=A027941(n-1)=Fibonacci(2n-1)-1.

%C Sum(k*T(n,k),k=0..floor(n/2))=A079309(n-1).

%C For the statistic "number of peaks in peak plateaux", see A143953.

%H David Callan, <a href="http://www.stat.wisc.edu/~callan/notes/">Kreweras's Narayana Number Identity Has a Simple Dyck Path Interpretation .</a>

%F The g.f. G=G(t,z) satisfies z(1-z)G^2 - (1-z+z^2-tz^2)G+1-z = 0 (for the explicit form of G see the Maple program).

%F The trivariate g.f. g=g(x,y,z) of Dyck paths with respect to number of peak plateaux, number of peaks in the peak plateaux and semilength, marked, by x, y and z, respectively satisfies g=1+zg[g+xyz/(1-yz)-z/(1-z)].

%F T(n,k) = Sum_{r=1..n} Narayana(n-r,k)*binomial(2n-r-k,r-k) where Narayana(n,k) := binomial(n,k)*binomial(n,k-1)/n is the Narayana number A001263. - _David Callan_, Oct 31 2008

%e T(3,1)=4 because we have UD(UUDD), (UUDD)UD, (UUDUDD) and U(UUDD)D (the peak plateaux are shown between parentheses).

%e The triangle starts:

%e 1;

%e 1;

%e 1,1;

%e 1,4;

%e 1,12,1;

%e 1,33,8;

%e 1,88,42,1;

%p C:=proc(z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: G:=(1-z)*C(z*(1-z)^2/(1-z+z^2-t*z^2)^2)/(1-z+z^2-t*z^2): Gser:=simplify(series(G,z= 0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

%Y Cf. A000045, A000108, A027941, A079309, A143953.

%K nonn,tabf

%O 0,6

%A _Emeric Deutsch_, Oct 10 2008

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 06:15 EDT 2024. Contains 371265 sequences. (Running on oeis4.)