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A101482 Column 1 of triangular matrix T=A101479, in which row n equals row (n-1) of T^(n-1) followed by '1'. 8
1, 1, 2, 9, 70, 795, 11961, 224504, 5051866, 132523155, 3969912160, 133678842902, 4997280555576, 205320100093953, 9195224163850830, 445775353262707365, 23255990676521697670, 1299028117862237432959, 77348967890083608924045 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 17 00:35 EDT 2021. Contains 348048 sequences. (Running on oeis4.)