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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).
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%I #16 May 02 2022 10:24:22

%S 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 1400,840,168,8,9,252,1764,4410,4410,1764,252,9,10,360,3360,11760,

%U 17640,11760,3360,360,10,11,495,5940,27720,58212,58212,27720,5940,495,11,12

%N 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 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 For double rises only above the x-axis see A118964.

%C This is the triangle of Narayana with row n multiplied by n + 1. - _Peter Luschny_, May 02 2022

%F T(n,1) = n(n^2 - 1)/2 (A027480).

%F T(n,2) = (n+1)n(n-1)^2*(n-2)/12 (A027789).

%F T(n,k) = ((n+1)/n)*binomial(n,k)*binomial(n,k+1).

%F Sum_{k>=0} k*T(n,k) = (2n-1)!/(n!(n-2)!) (A000917).

%F 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 Row n is given by seq(binomial(n, k)*binomial(n+2, n+1-k), k=0..n). - _Zerinvary Lajos_, Nov 03 2006

%F T(n,k)/(n+1) = A001263(n,k). - _Peter Luschny_, May 02 2022

%e T(3,2)=4 because we have uuuddd, duuudd, dduuud and ddduuu.

%e Triangle begins:

%e 2;

%e 3, 3;

%e 4, 12, 4;

%e 5, 30, 30, 5;

%e 6, 60, 120, 60, 6;

%e 7, 105, 350, 350, 105, 7;

%e 8, 168, 840, 1400, 840, 168, 8;

%e 9, 252, 1764, 4410, 4410, 1764, 252, 9;

%p 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 for n from 0 to 10 do seq(binomial(n,i)*binomial(n+2,n+1-i), i=0..n ); od; # _Zerinvary Lajos_, Nov 03 2006

%Y Cf. A000917, A000984 (row sums), A027480, A027789, A118964, A001263.

%K nonn,tabl

%O 1,1

%A _Emeric Deutsch_, May 07 2006