

A210697


Triangle read by rows, arising in study of alternatingsign matrices.


2



1, 1, 1, 2, 5, 2, 9, 36, 36, 9, 90, 495, 855, 495, 90, 2025, 14175, 34830, 34830, 14175, 2025, 102060, 867510, 2776032, 4082400, 2776032, 867510, 102060
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OFFSET

1,4


COMMENTS

See Mills et al., pp. 353354 and 359 for precise definition. As of 1983 no formula was known for these numbers.
These are the values of a bivariate generating function for the ASMs by numbers of entries equal to 1 and by position of 1 in the first row (see Example section). Here weight x=3 is chosen, giving a decomposition of the 3enumeration of the n X n ASMs.
As a triangle of coefficients of polynomials, A210697 has interesting properties relating the (2n+1)th row and the nth row (see Mills et al., p. 359).


LINKS

Table of n, a(n) for n=1..28.
W. H. Mills, David P Robbins, Howard Rumsey Jr., Alternating sign matrices and descending plane partitions J. Combin. Theory Ser. A 34 (1983), no. 3, 340359. MR0700040 (85b:05013). See p. 359.


EXAMPLE

The bivariate g.f. as a table of polynomials.
(degree of x is the count of 1 entries in the ASM)
Setting x=k gives the kenumeration of the ASMs
n
1  1
2  1, 1
3  2, 2+x, 2
4  6+x, 6+7*x+x^2, 6+7*x+x^2, 6+x
5  24 + 16*x + 2*x^2, 24 + 52*x + 26*x^2 + 3*x^3, 24 + 64*x + 38*x^2 +
 8*x^3 + x^4, 24 + 52*x + 26*x^2 + 3*x^3, 24 + 16*x + 2*x^2
...
Triangle begins:
n
1  1
2  1 1
3  2 5 2
4  9 36 36 9
5  90 495 855 495 90
6  2025 14175 34830 34830 14175 2025
...


CROSSREFS

A048601 is the version for x=1.
As for A048601, the row sums A059477 are equal to the first column, shifted by one.
Sequence in context: A272679 A119764 A178310 * A085020 A276609 A292591
Adjacent sequences: A210694 A210695 A210696 * A210698 A210699 A210700


KEYWORD

nonn,tabl,more


AUTHOR

N. J. A. Sloane, Mar 30 2012


EXTENSIONS

More terms, definitions and examples by Olivier Gérard, Apr 02 2015


STATUS

approved



