A101482 Column 1 of triangular matrix T=A101479, in which row n equals row (n-1) of T^(n-1) followed by '1'.

1, 1, 2, 9, 70, 795, 11961, 224504, 5051866, 132523155, 3969912160, 133678842902, 4997280555576, 205320100093953, 9195224163850830, 445775353262707365, 23255990676521697670, 1299028117862237432959, 77348967890083608924045

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

Links

Table of n, a(n) for n=0..18.

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

Cf. A101479, A101481, A101483, A136730.

Sequence in context: A300014 A108522 A014500 * A099717 A322772 A177450

Adjacent sequences: A101479 A101480 A101481 * A101483 A101484 A101485

Keyword

nonn

Author

Paul D. Hanna, Jan 21 2005

Status

approved