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%I #3 Mar 30 2012 17:35:59
%S 1,1,2,4,1,9,4,1,23,13,5,1,65,41,19,6,1,197,131,67,26,7,1,626,428,232,
%T 101,34,8,1,2056,1429,804,376,144,43,9,1,6918,4861,2806,1377,573,197,
%U 53,10,1,23714,16795,9878,5017,2211,834,261,64,11,1,82500,58785,35072
%N Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having leftmost valley at altitude k (if path has no valleys, then this altitude is considered to be 0).
%C Row sums are the Catalan numbers (A000108) Column 0 is A014137 (partial sums of Catalan numbers). Column 1 is A001453 (Catalan numbers -1).
%F G.f.=(1-z+zC-tzC)/[(1-z)(1-tzC)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%e Triangle starts:
%e 1;
%e 1;
%e 2;
%e 4,1;
%e 9,4,1;
%e 23,13,5,1;
%e 65,41,19,6,1;
%e T(4,1)=4 because we have UU(DU)DDUD, UU(DU)DUDD, UU(DU)UDDD and UUUD(DU)DD, where U=(1,1), D=(1,-1); the first valleys, all at altitude 1, are shown between parentheses.
%Y Cf. A000108, A014137, A001453.
%K nonn,tabf
%O 0,3
%A _Emeric Deutsch_, Aug 30 2004
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