OFFSET
0,3
COMMENTS
A generalized Catalan number triangle.
An alternative construction of this triangle. Begin with the Pascal triangle array, written as:
1 1 1 1 1 1 ...
1 2 3 4 5 6 ...
1 3 6 10 15 21 ...
1 4 10 20 35 56 ...
1 5 15 35 70 126 ...
...
For each row r (r >= 0) in the above array, construct a triangle U(r) by applying the operation H defined below.
Then the r-th diagonal from the right in the new triangle is given by the row sums of U(r).
To define H, let us use row r=2, {1 3 6 10 15 ...}, as an illustration.
To get the 4th entry, take the first 4 terms of the row, reverse them and write them under the first 4 terms:
A: 1 3 6 10
B: 10 6 3 1
and form a new row C by beginning with 1 and iterating the map C' = C*B/A until we reach 1:
C: 1 10 20 10 1
E.g., 20 = (6 *10) / 3.
The sum of the terms {1 10 20 10 1} is 42, which is the 4th entry in the r=2 diagonal of the new triangle.
The full triangle U(2) begins
1
1 1
1 3 1
1 6 6 1
1 10 20 10 1
...
(this is the Narayana triangle A001263)
and the row sums are the Catalan numbers, which give our r=2 diagonal.
LINKS
N. J. A. Sloane, First 30 rows, flattened
FORMULA
Comment from N. J. A. Sloane, Sep 07 2006: (Start)
The n-th entry in the r-th diagonal from the right (r >= 0, n >= 1) is given by the quotient:
Sum_{k=1..n} Product_{i=0..r-1} binomial(n+r-2, k-1+i)
------------------------------------------------------
Product_{i=1..r-1} binomial(n+r-2, i)
(End)
EXAMPLE
The first few rows of the triangle are:
1
1 2
1 2 3
1 2 4 4
1 2 5 8 5
1 2 6 14 16 6
1 2 7 22 42 32 7
1 2 8 32 92 132 64 8
1 2 9 44 177 422 429 128 9
1 2 10 58 310 1122 2074 1430 256 10
...
MAPLE
g:=proc(n, p) local k, i; 1 + add( mul( binomial(n+p-1, p+i) / binomial(n+p-1, i), i=0..k-1 ), k=1..n); end; (N. J. A. Sloane, based on the formula from Hsueh-Hsing Hung)
f:=proc(n, r) local k, b, i; b:=binomial; add( mul( b(n+r-2, k-1+i), i=0..r-1)/ mul( b(n+r-2, i), i=1..r-1), k=1..n); end; M:=30; for j from 0 to M do lprint(seq(f(i, j+1-i), i=1..j+1)); od; # N. J. A. Sloane
MATHEMATICA
rows = 11; t[n_, p_] := 1 + Sum[Product[ Binomial[ n+p-1, p+i] / Binomial[ n+p-1, i], {i, 0, k-1}], {k, 1, n}]; Flatten[ Table[ t[p, n-p], {n, 0, rows}, {p, 0, n}]](* Jean-François Alcover, Nov 18 2011, after Maple *)
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Feb 26 2006
EXTENSIONS
One entry corrected by Hsueh-Hsing Hung (hhh(AT)mail.nhcue.edu.tw), Sep 06 2006
Edited and extended by N. J. A. Sloane, Sep 07 2006
Simpler formula provided by Hsueh-Hsing Hung (hhh(AT)mail.nhcue.edu.tw), Sep 08 2006, which is now taken as the definition of this triangle
Edited by Jon E. Schoenfield, Dec 12 2015
STATUS
approved