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Number of symmetric nonnegative integer 7 X 7 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.
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%I #16 Jan 15 2017 08:02:07

%S 1,7,31,105,300,756,1732,3676,7330,13870,25102,43714,73612,120340,

%T 191620,298012,453739,677677,994565,1436435,2044328,2870296,3979768,

%U 5454280,7394660,9924668,13195196,17389028,22726280,29470520,37935704,48493928,61584149

%N Number of symmetric nonnegative integer 7 X 7 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.

%H Colin Barker, <a href="/A054497/b054497.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (7,-18,14,25,-63,36,36,-63,25,14,-18,7,-1).

%F G.f.: 1 / ((1-x)^7 * (1-x^2)^3).

%F a(0)=1, a(1)=7, a(2)=31, a(3)=105, a(4)=300, a(5)=756, a(6)=1732, a(7)=3676, a(8)=7330, a(9)=13870, a(10)=25102, a(11)=43714, a(12)=73612, a(n) = 7*a(n-1) - 18*a(n-2) + 14*a(n-3) + 25*a(n-4) - 63*a(n-5) + 36*a(n-6) + 36*a(n-7) - 63*a(n-8) + 25*a(n-9) + 14*a(n-10) - 18*a(n-11) + 7*a(n-12) - a(n-13). - _Harvey P. Dale_, Feb 03 2012

%F a(n) = ((2835*(4017+79*(-1)^n) + 18*(1513217+4095*(-1)^n)*n + 90*(284609+63*(-1)^n)*n^2 + 12771104*n^3 + 3787056*n^4 + 701400*n^5 + 81900*n^6 + 5856*n^7 + 234*n^8 + 4*n^9)) / 11612160. - _Colin Barker_, Jan 15 2017

%t CoefficientList[Series[1/((1-x)^7(1-x^2)^3),{x,0,30}],x] (* _Harvey P. Dale_, Feb 03 2012 *)

%o (PARI) Vec(1 / ((1-x)^7*(1-x^2)^3) + O(x^40)) \\ _Colin Barker_, Jan 15 2017

%Y Cf. A001753, A038163.

%K nonn,easy

%O 0,2

%A _Vladeta Jovovic_, May 14 2000