OFFSET
0,3
COMMENTS
Number of Dyck paths whose ascent lengths are exactly {n, n-1, .. 1}, for example the a(2) = 2 paths are uududd and uuddud. - David Scambler, May 30 2012
FORMULA
Equals column 0 of array A136730.
EXAMPLE
This sequence can also be generated in the following manner.
Start a table with the all 1's sequence in row 0; from then on, row n+1 can be formed from row n by dropping the initial n terms of row n and taking partial sums of the remaining terms to obtain row n+1.
The following table (A136730) illustrates this method:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
[1], 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...;
[2, 5], 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, ...;
[9, 23, 43], 70, 105, 149, 203, 268, 345, 435, 539, 658, ...;
[70, 175, 324, 527], 795, 1140, 1575, 2114, 2772, 3565, ...;
[795, 1935, 3510, 5624, 8396], 11961, 16471, 22096, 29025, ...;
[11961, 28432, 50528, 79553, 117020, 164672], 224504, ...; ...
In the above table, drop the initial n terms in row n (enclosed in square brackets) and then take partial sums to obtain row n+1 for n>=0;
this sequence then forms the first column of the resultant table.
Note: column k of the above table equals column 1 of matrix power T^(k+1) where T=A101479, for k>=0.
PROG
(PARI) {a(n)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(A^(i-1))[i-1, j]); )); A=B); return(A[n+1, 1])}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 21 2005
STATUS
approved