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A210697 Triangle read by rows, arising in study of alternating-sign 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

See Mills et al., pp. 353-354 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 3-enumeration 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 n-th 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, 340--359. 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 k-enumeration 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

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Last modified May 21 05:37 EDT 2022. Contains 353889 sequences. (Running on oeis4.)