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Number of symmetric nonnegative integer 8 X 8 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.
4

%I #17 Jan 15 2017 08:25:27

%S 1,4,16,44,116,260,560,1100,2090,3740,6512,10868,17732,28028,43472,

%T 65780,97955,143000,205920,291720,408408,563992,770848,1041352,

%U 1394068,1847560,2428960,3165400,4095640,5258440,6708064,8498776,10705189,13401916,16689904

%N Number of symmetric nonnegative integer 8 X 8 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.

%D Y. Teranishi, Linear Diophantine equations and invariant theory of matrices, in Commutative algebra and combinatorics (Kyoto, 1985), pp. 259-275, Adv. Stud. Pure Math., 11, North-Holland, Amsterdam, 1987. (See p. 273.)

%H Alois P. Heinz, <a href="/A054498/b054498.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-20,20,36,-64,-20,90,-20,-64,36,20,-20,0,4,-1).

%F G.f.: 1 / ((1-x)^4 * (1-x^2)^6).

%F a(n) = ((8+n)*(2835*(1739+309*(-1)^n) + 576*(15259+1029*(-1)^n)*n + 36*(166171+3717*(-1)^n)*n^2 + 448*(4661+27*(-1)^n)*n^3 + 14*(29749+27*(-1)^n)*n^4 + 49280*n^5 + 3416*n^6 + 128*n^7 + 2*n^8)) / 46448640. - _Colin Barker_, Jan 15 2017

%o (PARI) Vec(1 / ((1-x)^4*(1-x^2)^6) + O(x^40)) \\ _Colin Barker_, Jan 15 2017

%Y Cf. A001753, A038163.

%Y Column k=4 of A210391. - _Alois P. Heinz_, Mar 22 2012

%K easy,nonn

%O 0,2

%A _Vladeta Jovovic_, May 14 2000