A210697 Triangle read by rows, arising in study of alternating-sign matrices.

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

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 A358309 A276609

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