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%I #46 Jan 18 2024 09:52:25
%S 1,5,16,40,86,166,296,496,791,1211,1792,2576,3612,4956,6672,8832,
%T 11517,14817,18832,23672,29458,36322,44408,53872,64883,77623,92288,
%U 109088,128248,150008,174624,202368,233529
%N Expansion of 1/((1+x)*(1-x)^6).
%C Number of symmetric nonnegative integer 5 X 5 matrices with sum of elements equal to 4*n under action of dihedral group D_4.
%C a(n) = A108561(n+6,n) for n>0. - _Reinhard Zumkeller_, Jun 10 2005
%H Vincenzo Librandi, <a href="/A001753/b001753.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,5,5,-9,5,-1)
%F a(n) = Sum{k=0..n} (-1)^(n-k)*binomial(k+5, 5); a(n) = (4*n^5 + 70*n^4 + 460*n^3 + 1400*n^2 + 1936*n + 945)/960 + (-1)^n/64. - _Paul Barry_, Jul 01 2003
%F a(n) = a(n-2) + (n*(n + 1)*(n + 2)*(n - 1))/24, a(1) = 0, a(2) = 1; (15*(-1)^n - 15*(-1)^(2*n) + 96*n - 160*(-1)^(2*n)*n + 200*n^2 - 200*(-1)^(2*n)*n^2 + 140*n^3 - 80*(-1)^(2*n)*n^3 + 40*n^4 - 10*(-1)^(2*n)*n^4 + 4*n^5)/960. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004
%F a(n) + a(n+1) = A000389(n+6). - _R. J. Mathar_, Mar 14 2011
%e There are 5 symmetric nonnegative integer 5 X 5 matrices with sum of elements equal to 4 under action of D_4:
%e [1 0 0 0 1] [0 0 1 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0]
%e [0 0 0 0 0] [0 0 0 0 0] [0 1 0 1 0] [0 0 1 0 0] [0 0 0 0 0]
%e [0 0 0 0 0] [1 0 0 0 1] [0 0 0 0 0] [0 1 0 1 0] [0 0 4 0 0]
%e [0 0 0 0 0] [0 0 0 0 0] [0 1 0 1 0] [0 0 1 0 0] [0 0 0 0 0]
%e [1 0 0 0 1] [0 0 1 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0].
%t CoefficientList[Series[1/((1+x)*(1-x)^6), {x, 0, 50}], x] (* _G. C. Greubel_, Nov 22 2017 *)
%t LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,5,16,40,86,166,296},40] (* _Harvey P. Dale_, Jun 05 2021 *)
%o (Magma) [(4*n^5+70*n^4+460*n^3+1400*n^2+1936*n+945)/960+(-1)^n/64: n in [0..40]]; // _Vincenzo Librandi_, Aug 15 2011
%o (PARI) a(n)=(4*n^5+70*n^4+460*n^3+1400*n^2+1936*n)\/960+1 \\ _Charles R Greathouse IV_, Apr 17 2012
%Y Cf. A000217, A002620, A008804, A038163, A054343, A001769 (partial sums), A001752 (first differences), A169793 (binomial transf).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Comment and example from _Vladeta Jovovic_, May 14 2000
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