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%I #22 Sep 08 2022 08:45:49
%S 0,1,4160,798255,33562624,610390625,6530486976,48444916975,
%T 274878955520,1270935305649,5000005000000,17261365815551,
%U 53496620605440,151437584670385,396857439333824,973097619609375,2251799947902976
%N a(n) = n^7*(n^6 + 1)/2.
%C Number of unoriented rows of length 13 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=4160, there are 2^13=8192 oriented arrangements of two colors. Of these, 2^7=128 are achiral. That leaves (8192-128)/2=4032 chiral pairs. Adding achiral and chiral, we get 4160. - _Robert A. Russell_, Nov 13 2018
%H Vincenzo Librandi, <a href="/A168663/b168663.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).
%F From _G. C. Greubel_, Jul 28 2016: (Start)
%F G.f.: x*(1 + 4146*x + 740106*x^2 + 22765250*x^3 + 211641855*x^4 + 752814348*x^5 + 1137578988*x^6 + 752814348*x^7 + 211641855*x^8 + 22765250*x^9 + 740106*x^10 + 4146*x^11 + x^12)/(1 - x)^14.
%F E.g.f.: (1/2)*x*(2 + 4158*x + 261926*x^2 + 2532880*x^3 + 7508641*x^4 + 9321333*x^5 + 5715425*x^6 + 1899612*x^7 + 359502*x^8 + 39325*x^9 + 2431*x^10 + 78*x^11 + x^12)*exp(x). (End)
%F From _Robert A. Russell_, Nov 13 2018: (Start)
%F a(n) = (A010801(n) + A001015(n)) / 2 = (n^13 + n^7) / 2.
%F G.f.: (Sum_{j=1..13} S2(13,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
%F G.f.: x*Sum_{k=0..12} A145882(13,k) * x^k / (1-x)^14.
%F E.g.f.: (Sum_{k=1..13} S2(13,k)*x^k + Sum_{k=1..7} S2(7,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
%F For n>13, a(n) = Sum_{j=1..14} -binomial(j-15,j) * a(n-j). (End)
%t Table[n^7(n^6+1)/2,{n,0,20}] (* _Harvey P. Dale_, Jan 20 2013 *)
%o (Magma) [n^7*(n^6+1)/2: n in [0..20]]; // _Vincenzo Librandi_, Aug 28 2011
%o (PARI) a(n)=n^7*(n^6+1)/2 \\ _Charles R Greathouse IV_, Jul 28 2016
%Y Row 13 of A277504.
%Y Cf. A010801 (oriented), A001015 (achiral).
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Dec 11 2009
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