Search: id:A118963
Results 1-1 of 1 results found.

%I A118963
%S A118963 2,3,3,4,12,4,5,30,30,5,6,60,120,60,6,7,105,350,350,105,7,8,168,840,
%T A118963 1400,840,168,8,9,252,1764,4410,4410,1764,252,9,10,360,3360,11760,17640,
%U A118963 11760,3360,360,10,11,495,5940,27720,58212,58212,27720,5940,495,11,12
%N A118963 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength 
               n that have k double rises (n>=1,k>=0).
%C A118963 A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting 
               at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,
               -1); a double rise in a Grand Dyck path is an occurrence of uu in 
               the path.
%C A118963 Row sums are the central binomial coefficients (A000984). T(n,0)=n+1. 
               T(n,1)=n(n^2-1)/2 (A027480). T(n,2)=(n+1)n(n-1)^2*(n-2)/12 (A027789). 
               Sum(k*T(n,k),k>=0)=(2n-1)!/[n!(n-2)! ] (A000917). For double rises 
               only above the x-axis, see A118964.
%F A118963 T(n,k)=[(n+1)/n]binomial(n,k)binomial(n,k+1). G.f.=G(t,z)=(1+r)^2/(1-tr^2)-1, 
               where r=r(t,z) is the Narayana function, defined by (1+r)(1+tr)z=r, 
               r(t,0)=0. More generally, the g.f. H=H(t,s,u,z), where t,s and u 
               mark double rises above, below and on the x-axis, respectively, is 
               H=[1+r(s,z)]/[1-z(1+tr(t,z))(1+ur(s,z))].
%F A118963 Row n is given by seq(binomial(n,i)*binomial(n+2,n+1-i), i=0..n ). - 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 03 2006
%e A118963 T(3,2)=4 because we have uuuddd, duuudd, dduuud and ddduuu.
%e A118963 Triangle begins:
%e A118963 2
%e A118963 3, 3
%e A118963 4, 12, 4
%e A118963 5, 30, 30, 5
%e A118963 6, 60, 120, 60, 6
%e A118963 7, 105, 350, 350, 105, 7
%e A118963 8, 168, 840, 1400, 840, 168, 8
%e A118963 9, 252, 1764, 4410, 4410, 1764, 252, 9
%p A118963 r:=(1-z-t*z-sqrt(z^2*t^2-2*z^2*t-2*z*t+z^2-2*z+1))/2/t/z: G:=(1+r)^2/
               (1-t*r^2)-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 
               do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 11 do seq(coeff(P[n],
               t,j),j=0..n-1) od; # yields sequence in triangular form
%p A118963 for n from 0 to 10 do seq(binomial(n,i)*binomial(n+2,n+1-i), i=0..n ); 
               od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 03 2006
%Y A118963 Cf. A000984, A027480, A027789, A000917, A118964.
%Y A118963 Sequence in context: A130743 A111574 A128744 this_sequence A127641 A106821 
               A065863
%Y A118963 Adjacent sequences: A118960 A118961 A118962 this_sequence A118964 A118965 
               A118966
%K A118963 nonn,tabl
%O A118963 1,1
%A A118963 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 07 2006

Search completed in 0.001 seconds