%I #4 Mar 30 2012 17:36:07
%S 1,1,1,1,3,1,1,8,4,1,1,22,13,5,1,1,64,41,19,6,1,1,196,131,67,26,7,1,1,
%T 625,428,232,101,34,8,1,1,2055,1429,804,376,144,43,9,1,1,6917,4861,
%U 2806,1377,573,197,53,10,1,1,23713,16795,9878,5017,2211,834,261,64,11,1,1
%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having length of second ascent equal to k (0<=k<=n-1).
%C Column 1 yields A014138, column 2 yields A001453, column 3 yields A114277. Row sums are the Catalan numbers (A000108).
%F T(n, k)=(k+1)*sum(binomial(2*n-k+1-2*j, n-j+1)/(2*n-k-2*j+1), j=1..n-k) if 1<=k<=n-1; T(n, 0)=1. G.f. = (1-tz)/[(1-z)(1-tzC)]-1 where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%e T(4,2)=4 because we have UD(UU)DDUD, UD(UU)DUDD, UUD(UU)DDD and UUDD(UU)DD (second ascent shown between parentheses).
%p T:=proc(n,k) if k=0 then 1 elif k<=n-1 then (k+1)*sum(binomial(2*n-k+1-2*j,n-j+1)/(2*n-k-2*j+1),j=1..n-k) else 0 fi end: for n from 1 to 12 do seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form
%Y Cf. A000108, A014138, A001453, A114277.
%K nonn,tabl
%O 1,5
%A _Emeric Deutsch_, Nov 20 2005