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Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (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)).
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%I #10 Jun 13 2017 23:46:55

%S 1,2,5,1,14,6,42,27,1,132,110,10,429,429,65,1,1430,1638,350,14,4862,

%T 6188,1700,119,1,16796,23256,7752,798,18,58786,87210,33915,4655,189,1,

%U 208012,326876,144210,24794,1518,22,742900,1225785,600875,123970,10350

%N Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (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)).

%C Row n contains 1+floor(n/2) terms. Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) (the Catalan numbers). T(n,1)=A003517(n). T(n,2)=A003519(n). Sum(k*T(n,k),k>=0)=A008549(n-1). For both downward and upward crossings, see A118920.

%C Eigenvector is defined by: A119243(n) = Sum_{k=0..[n\2]} T(n,k)*A119243(k). This triangle is closely related to triangle A119245. - _Paul D. Hanna_, May 10 2006

%C Column k is the sum of columns 2k and 2k+1 of A039599. - _Philippe Deléham_, Nov 11 2008

%F T(n,k)=(2k+1)binomial(2n+2,n-2k)/(n+1). G.f.=G(t,z)=C^2/(1-tz^2*C^4), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

%F T(n,k)=A039599(n,2k)+A039599(n,2k+1). - _Philippe Deléham_, Nov 11 2008

%e T(3,1)=6 because we have ud\dudu,ud\dduu,udud\du,uudd\du,ud\duud and duud\du (the downward crossings of the x-axis are shown by a back-slash \).

%e Triangle starts:

%e 1;

%e 2;

%e 5,1;

%e 14,6;

%e 42,27,1;

%e 132,110,10;

%p T:=(n,k)->(2*k+1)*binomial(2*n+2,n-2*k)/(n+1): for n from 0 to 13 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

%o (PARI) T(n,k)=if(n<2*k || k<0,0,(2*k+1)*binomial(2*n+2,n-2*k)/(n+1)) - _Paul D. Hanna_, May 10 2006

%Y Cf. A000984, A000108, A003517, A003519, A008549, A118920.

%Y Cf. A119243 (eigenvector), A119245 (variant).

%K nonn,tabf

%O 0,2

%A _Emeric Deutsch_, May 06 2006