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A143949 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).

%I A143949 #5 Jun 23 2017 07:58:12
%S A143949 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 A143949 155,94,55,13,11,0,1,329,502,335,154,80,16,13,0,1,1101,1701,1110,580,
%U A143949 226,109,19,15,0,1,3761,5820,3865,1960,898,310,142,22,17,0,1,13035,20251
%N A143949 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 A143949 Row sums are the Catalan numbers (A000108).
%C A143949 T(0,n)=A033297(n).
%C A143949 Sum(k*T(n,k),k=0..n)=A000957(n+2) (the Fine numbers).
%C A143949 The case of even-length descents to ground level is considered in A111301.
%F A143949 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 A143949 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 A143949 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 A143949 Triangle starts:
%e A143949 1;
%e A143949 0,1;
%e A143949 1,0,1;
%e A143949 1,3,0,1;
%e A143949 4,4,5,0,1;
%e A143949 10,17,7,7,0,1;
%p A143949 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 A143949 Cf. A000108, A033297, A000957, A111301.
%K A143949 nonn,tabl
%O A143949 0,8
%A A143949 _Emeric Deutsch_, Oct 05 2008