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%I #32 Sep 08 2022 08:45:49
%S 0,1,8256,2392578,134225920,3051796875,39182222016,339111948196,
%T 2199024304128,11438398618965,50000005000000,189874926535206,
%U 641959250190336,1968688224223903,5556003465485760,14596463098125000,36028797153181696,84188913484869801
%N a(n) = n^7*(n^7 + 1)/2.
%C Number of unoriented rows of length 14 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)=8256, there are 2^14=16384 oriented arrangements of two colors. Of these, 2^7=128 are achiral. That leaves (16384-128)/2=8128 chiral pairs. Adding achiral and chiral, we get 8256. - _Robert A. Russell_, Nov 13 2018
%H Vincenzo Librandi, <a href="/A168664/b168664.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005, -3003,1365,-455,105,-15,1).
%F From _Wesley Ivan Hurt_, Oct 30 2014: (Start)
%F G.f.: (x + 8241*x^2 + 2268843*x^3 + 99203675*x^4 + 1285873650*x^5 + 6421633938*x^6 + 13985577438*x^7 + 13985598654*x^8 + 6421628925*x^9 + 1285868525*x^10 + 99207111*x^11 + 2268471*x^12 + 8128*x^13)/(1 - x)^15.
%F a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15).
%F a(n) = n^7*(n^7 + 1)/2 = A000217(A001015(n)). (End)
%F From _Robert A. Russell_, Nov 13 2018: (Start)
%F a(n) = (A010802(n) + A001015(n)) / 2 = (n^14 + n^7) / 2.
%F G.f.: (Sum_{j=1..14} S2(14,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..13} A145882(14,k) * x^k / (1-x)^15.
%F E.g.f.: (Sum_{k=1..14} S2(14,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>14, a(n) = Sum_{j=1..15} -binomial(j-16,j) * a(n-j). (End)
%F E.g.f.: x*(2+8254*x +789271*x^2 +10392095*x^3 +40075175*x^4 +63436394*x^5 +49329281*x^6 +20912320*x^7 +5135130*x^8 +752752*x^9 + 66066*x^10 +3367*x^11 +91*x^12 +x^13)*exp(x)/2. - _G. C. Greubel_, Nov 15 2018
%p A168664:=n->n^7*(n^7+1)/2: seq(A168664(n), n=0..20); # _Wesley Ivan Hurt_, Oct 30 2014
%t f[n_]:=Module[{c=n^7},c (c+1)/2]; f/@Range[0,30] (* _Harvey P. Dale_, Mar 19 2011 *)
%o (Magma) [n^7*(n^7+1)/2: n in [0..20]]; // _Vincenzo Librandi_, Aug 28 2011
%o (PARI) a(n)=n^7*(n^7+1)/2 \\ _Charles R Greathouse IV_, Jul 28 2016
%o (Sage) [n^7*(1 + n^7)/2 for n in range(40)] # _G. C. Greubel_, Nov 15 2018
%o (GAP) List([0..30], n -> n^7*(1 + n^7)/2); # _G. C. Greubel_, Nov 15 2018
%Y Cf. A001015 (Seventh Powers: n^7), A000217 (Triangular Numbers).
%Y Row 14 of A277504.
%Y Cf. A010802 (oriented), A001015 (achiral).
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Dec 11 2009
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