A114276 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).
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, 625, 428, 232, 101, 34, 8, 1, 1, 2055, 1429, 804, 376, 144, 43, 9, 1, 1, 6917, 4861, 2806, 1377, 573, 197, 53, 10, 1, 1, 23713, 16795, 9878, 5017, 2211, 834, 261, 64, 11, 1, 1
Offset: 1
Examples
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).
Programs
-
Maple
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
Formula
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.
Comments