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Number of symmetric nonnegative integer 9 X 9 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.
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%I #11 Jan 15 2017 08:14:40

%S 1,9,51,219,786,2466,6974,18126,43929,100321,217683,451707,901128,

%T 1735752,3239928,5878328,10393902,17950878,30340682,50273658,81787476,

%U 130811124,205935756,319456044,488764246,738197766,1101468114,1624826306,2371158504,3425244456

%N Number of symmetric nonnegative integer 9 X 9 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.

%H Colin Barker, <a href="/A054549/b054549.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (9,-30,30,75,-243,152,360,-690,130,780,-780,-130,690,-360,-152,243,-75,-30,30,-9,1).

%F G.f.: 1 / ((1-x)^9 * (1-x^2)^6).

%F a(n) = ((85135050*(256719+5425*(-1)^n) + 1890*(30592018355+141137997*(-1)^n)*n + 9*(7020005494399+6456074625*(-1)^n)*n^2 + 15288*(2534162507+393525*(-1)^n)*n^3 + 455*(33274853083+654885*(-1)^n)*n^4 + 126126*(31995443+45*(-1)^n)*n^5 + 763179408992*n^6 + 104737240608*n^7 + 10538322080*n^8 + 777020244*n^9 + 41479438*n^10 + 1559376*n^11 + 39130*n^12 + 588*n^13 + 4*n^14)) / 22317642547200. - _Colin Barker_, Jan 15 2017

%o (PARI) Vec(1 / ((1-x)^9*(1-x^2)^6) + O(x^40)) \\ _Colin Barker_, Jan 15 2017

%Y Cf. A001753, A038163.

%K easy,nonn

%O 0,2

%A _Vladeta Jovovic_, May 14 2000