A101975 Triangle read by rows: number of Dyck paths of semilength n with k peaks after the first return (0 <= k < n).

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, 6979, 2917, 578, 46, 1

Offset: 1

Author

Emeric Deutsch, Dec 22 2004

Keywords

• nonn
• tabl

Examples

			T(4,2)=4 because we have UD|(UD)U(UD)D, UD|U(UD)D(UD), UD|U(UD)(UD)D and
UUDD|(UD)(UD), where U=(1,1), D=(1,-1) (the two peaks after the first return | are shown between parentheses).
Triangle begins:
   1;
   1,  1;
   2,  2,  1;
   5,  4,  4,  1;
  14,  9, 11,  7,  1;
  42, 23, 27, 28, 11,  1;
  ...
		

References

• Emeric Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.

Crossrefs

Cf. A000108, A014137, A101974.

Programs

Maple

c:=n->binomial(2*n,n)/(n+1):T:=proc(n,k) if k=0 then c(n-1) elif k=1 then sum(c(i),i=0..n-2) else sum(c(j)*binomial(n-1-j,k-1)*binomial(n-1-j,k)/(n-1-j),j=0..n-2) fi end: for n from 1 to 11 do seq(T(n,k),k=0..n-1) od; # yields the sequence in triangular form

Formula

T(n,0) = c(n-1), T(n,1) = Sum_{i=0..n-2} c(i), T(n,k) = Sum_{j=0..n-2} c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j) for k >= 2, where c(i) = binomial(2i, i)/(i+1) (i=0, 1, ...) are the Catalan numbers (A000108).