|
| |
|
|
A212207
|
|
Triangle read by rows: coefficients of polynomials p_{n,n-1}(x) arising in enumeration of two-line arrays.
|
|
0
|
|
|
|
1, 1, 1, 1, 3, 2, 1, 6, 9, 4, 1, 10, 26, 25, 8, 1, 15, 60, 95, 65, 16, 1, 21, 120, 280, 309, 161, 32, 1, 28, 217, 700, 1113, 924, 385, 64, 1, 36, 364, 1554, 3346, 3948, 2596, 897, 128, 1, 45, 576, 3150, 8820, 13902, 12864, 6957, 2049, 256, 1, 55, 870, 5940
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
These polynomials are defined in Section 3 of Carlitz-Riordan (1971). Equation (3.14) claims to be a recurrence, which unfortunately I could not get to work. The coefficients of the polynomials A_n(x) = a_{n,n}(x) which appear in (3.14) are the Narayana numbers A001263.
|
|
|
LINKS
|
Table of n, a(n) for n=0..58.
Carlitz, L. and Riordan, John. Enumeration of some two-line arrays by extent. J. Combinatorial Theory Ser. A 10 1971 271--283. MR0274301(43 #66).
L. Carlitz and J. Riordan, Enumeration of some two-line arrays by extent, J. Combinatorial Theory Ser. A 10 1971 271-283 (MR274301 Review by Richard P. Stanley)
|
|
|
FORMULA
|
G.f.: N(x,y)/(1-N(x,y)^2), where N(x,y) is g.f. of Narayana numbers (A001263). -Vladimir Kruchinin, Apr 10 2018
|
|
|
EXAMPLE
|
Triangle begins:
1
1 1
1 3 2
1 6 9 4
1 10 26 25 8
...
|
|
|
PROG
|
(PARI) {T(n, k) = if( n < k || k < 0, 0, sum( j=0, k, binomial( n+1, k+1) * binomial( n+1, k-j) * if( j%2, -(n+1 +j-k), k+1)) / (n+1))} /* Michael Somos, Aug 22 2012 */
|
|
|
CROSSREFS
|
Cf. A001263.
Sequence in context: A227790 A181897 A337977 * A111049 A211955 A088617
Adjacent sequences: A212204 A212205 A212206 * A212208 A212209 A212210
|
|
|
KEYWORD
|
nonn,tabl
|
|
|
AUTHOR
|
N. J. A. Sloane, May 15 2012
|
|
|
STATUS
|
approved
|
| |
|
|
