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Triangle read by rows: T(n,k) is the number of n-Dyck paths containing k odd-length descents to ground level (0<=k<=n).
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%I #5 Jun 23 2017 07:58:12

%S 1,0,1,1,0,1,1,3,0,1,4,4,5,0,1,10,17,7,7,0,1,32,46,34,10,9,0,1,100,

%T 155,94,55,13,11,0,1,329,502,335,154,80,16,13,0,1,1101,1701,1110,580,

%U 226,109,19,15,0,1,3761,5820,3865,1960,898,310,142,22,17,0,1,13035,20251

%N Triangle read by rows: T(n,k) is the number of n-Dyck paths containing k odd-length descents to ground level (0<=k<=n).

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

%C T(0,n)=A033297(n).

%C Sum(k*T(n,k),k=0..n)=A000957(n+2) (the Fine numbers).

%C The case of even-length descents to ground level is considered in A111301.

%F G.f.: G(s,z) = 1/[1-z(t+zC)/(1-z^2*C^2)], where C = [1-sqrt(1-4z)]/(2z) is the Catalan function.

%F The trivariate g.f. H(t,s,z), where t (s) marks odd-length (even-length) descents to ground level and z marks semilength, is H=1/[1-z(t+szC)/(1-z^2*C^2)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

%e T(4,2) = 5 because we have U(D)U(D)UUDD, U(D)UUDDU(D), U(D)UUU(DDD), UUDDU(D)U(D) and UUU(DDD)U(D) (the odd-length descents to ground level are shown between parentheses).

%e Triangle starts:

%e 1;

%e 0,1;

%e 1,0,1;

%e 1,3,0,1;

%e 4,4,5,0,1;

%e 10,17,7,7,0,1;

%p C:=((1-sqrt(1-4*z))*1/2)/z: G:=1/(1-z*(t+z*C)/(1-z^2*C^2)): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(expand(coeff(Gser,z,n))) end do: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) end do; # yields sequence in triangular form

%Y Cf. A000108, A033297, A000957, A111301.

%K nonn,tabl

%O 0,8

%A _Emeric Deutsch_, Oct 05 2008