%I #30 Sep 08 2022 08:45:48

%S 0,1,72,1134,8320,39375,140616,412972,1050624,2394765,5005000,9750906,

%T 17926272,31388539,52725960,85455000,134250496,205211097,306162504,

%U 447001030,640080000,900641511,1247296072,1702552644,2293401600

%N a(n) = n^4*(n^3 + 1)/2.

%C Number of unoriented rows of length 7 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)=72, there are 2^7=128 oriented arrangements of two colors. Of these, 2^4=16 are achiral. That leaves (128-16)/2=56 chiral pairs. Adding achiral and chiral, we get 72. - _Robert A. Russell_, Nov 13 2018

%H Vincenzo Librandi and Bruno Berselli, <a href="/A168194/b168194.txt">Table of n, a(n) for n = 0..1000</a> (first 480 terms from Vincenzo Librandi).

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F G.f.: x*(1 + 64*x + 586*x^2 + 1208*x^3 + 605*x^4 + 56*x^5)/(1-x)^8. - _Colin Barker_, Apr 26 2012

%F From _Robert A. Russell_, Nov 13 2018: (Start)

%F a(n) = (A001015(n) + A000583(n)) / 2 = (n^7 + n^4) / 2.

%F G.f.: (Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..4} S2(4,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

%F G.f.: x*Sum_{k=0..6} A145882(7,k) * x^k / (1-x)^8.

%F E.g.f.: (Sum_{k=1..7} S2(7,k)*x^k + Sum_{k=1..4} S2(4,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

%F For n>7, a(n) = Sum_{j=1..8} -binomial(j-9,j) * a(n-j). (End)

%F E.g.f.: x*(2 +70*x +307*x^2 +351*x^3 +140*x^4 +21*x^5 +x^6)*exp(x)/2. - _G. C. Greubel_, Nov 14 2018

%t Table[(n^4 (n^3+1))/2,{n,0,40}] (* _Harvey P. Dale_, Apr 29 2011 *)

%o (Magma) [n^4*(n^3+1)/2: n in [0..50]]; // _Vincenzo Librandi_, Apr 25 2011

%o (PARI) vector(50, n, n--; n^4*(n^3+1)/2) \\ _G. C. Greubel_, Nov 14 2018

%o (Sage) [n^4*(n^3+1)/2 for n in (0..50)] # _G. C. Greubel_, Nov 14 2018

%Y Row 7 of A277504.

%Y Cf. A001015 (oriented), A000583 (achiral).

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Dec 11 2009