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

%I A210697 #18 Nov 25 2016 05:28:38
%S A210697 1,1,1,2,5,2,9,36,36,9,90,495,855,495,90,2025,14175,34830,34830,14175,
%T A210697 2025,102060,867510,2776032,4082400,2776032,867510,102060
%N A210697 Triangle read by rows, arising in study of alternating-sign matrices.
%C A210697 See Mills et al., pp. 353-354 and 359 for precise definition. As of 1983 no formula was known for these numbers.
%C A210697 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.
%C A210697 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).
%H A210697 W. H. Mills, David P Robbins, Howard Rumsey Jr., <a href="http://dx.doi.org/10.1016/0097-3165(83)90068-7">Alternating sign matrices and descending plane partitions</a> J. Combin. Theory Ser. A 34 (1983), no. 3, 340--359. MR0700040 (85b:05013). See p. 359.
%e A210697 The bivariate g.f. as a table of polynomials.
%e A210697 (degree of x is the count of -1 entries in the ASM)
%e A210697 Setting x=k gives the k-enumeration of the ASMs
%e A210697 n
%e A210697 1 | 1
%e A210697 2 | 1, 1
%e A210697 3 | 2, 2+x, 2
%e A210697 4 | 6+x, 6+7*x+x^2, 6+7*x+x^2, 6+x
%e A210697 5 | 24 + 16*x + 2*x^2, 24 + 52*x + 26*x^2 + 3*x^3, 24 + 64*x + 38*x^2 +
%e A210697   |      8*x^3 + x^4, 24 + 52*x + 26*x^2 + 3*x^3, 24 + 16*x + 2*x^2
%e A210697 ...
%e A210697 Triangle begins:
%e A210697 n
%e A210697 1 |    1
%e A210697 2 |    1     1
%e A210697 3 |    2     5     2
%e A210697 4 |    9    36    36     9
%e A210697 5 |   90   495   855   495    90
%e A210697 6 | 2025 14175 34830 34830 14175  2025
%e A210697 ...
%Y A210697 A048601 is the version for x=1.
%Y A210697 As for A048601, the row sums A059477 are equal to the first column, shifted by one.
%K A210697 nonn,tabl,more
%O A210697 1,4
%A A210697 _N. J. A. Sloane_, Mar 30 2012
%E A210697 More terms, definitions and examples by _Olivier Gérard_, Apr 02 2015