proposed
proposed
editing
1, 1, 1, 2, 2, 1, 5, 4, 4, 1, 14, 9, 11, 7, 1, 42, 23, 27, 28, 11, 1, 132, 65, 66, 87, 62, 16, 1, 429, 197, 170, 239, 250, 122, 22, 1, 1430, 626, 471, 627, 829, 630, 219, 29, 1, 4862, 2056, 1398, 1656, 2448, 2553, 1419, 366, 37, 1, 16796, 6918, 4381, 4554, 6803, 8813
proposed
editing
Triangle read by rows: number of Dyck paths of semilength n with k peaks after the first return (0<= k <n).
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
T(n, 0)=c(n-1), T(n, 1) = sum(c(i), i=0..n-2), T(n, k)= sum(c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j), j=0..n-2) for k>=2, where c(i)=binomial(2i, i)/(i+1) (i=0, 1, ...) are the Catalan numbers (A000108).
1;
1,1;
2,2,1;
5,4,4,1;
14,9,11,7,1;
42,23,27,28,11,1;
approved
Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 22 2004
T(4,2)=4 because we have UD|(UD)U(UD)D, UD|U(UD)D(UD), UD|U(UD)(UD)D, and
nonn,tabl,new
T(n,0)=c(n-1), T(n,1) = sum(c(i),i=0..n-2), T(n,k)= sum(c(j)*binomial(n-1-j,k-1)*binomial(n-1-j,k)/(n-1-j),j=0..n-2) for k>=2, where c(i)=binomial(2i,i)/(i+1) (i=0,1,...) are the Catalan numbers (A000108).
nonn,tabl,new