A168372 - OEIS

%I #20 Sep 08 2022 08:45:49

%S 0,1,272,9963,131584,978125,5042736,20185207,67125248,193739769,

%T 500050000,1179054371,2580014592,5302435333,10330792304,19222059375,

%U 34360262656,59294648177,99180589968,161345086939,256001600000

%N a(n) = n^5*(n^4 + 1)/2.

%C Number of unoriented rows of length 9 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)=272, there are 2^9=512 oriented arrangements of two colors. Of these, 2^5=32 are achiral. That leaves (512-32)/2=240 chiral pairs. Adding achiral and chiral, we get 272. - _Robert A. Russell_, Nov 13 2018

%H Vincenzo Librandi, <a href="/A168372/b168372.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

%F G.f.: x*(1 + 262*x + 7288*x^2 + 44074*x^3 + 78190*x^4 + 44074*x^5 + 7288*x^6 + 262*x^7 + x^8)/(1 - x)^10. - _G. C. Greubel_, Jul 19 2016

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

%F a(n) = (A001017(n) + A000584(n)) / 2 = (n^9 + n^5) / 2.

%F G.f.: (Sum_{j=1..9} S2(9,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..5} S2(5,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

%F G.f.: x*Sum_{k=0..8} A145882(9,k) * x^k / (1-x)^10.

%F E.g.f.: (Sum_{k=1..9} S2(9,k)*x^k + Sum_{k=1..5} S2(5,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

%F For n>9, a(n) = Sum_{j=1..10} -binomial(j-11,j) * a(n-j). (End)

%F E.g.f.: x*(2 +270*x +3050*x^2 +7780*x^3 +6952*x^4 +2646*x^5 +462*x^6 + 36*x^7 +x^8)*exp(x)/2. - _G. C. Greubel_, Nov 15 2018

%t Table[n^5*(n^4 + 1)/2, {n,0,50}] (* _G. C. Greubel_, Jul 19 2016 *)

%o (Magma) [n^5*(n^4+1)/2: n in [0..30]]; // _Vincenzo Librandi_, Aug 28 2011

%o (PARI) a(n)=n^5*(n^4+1)/2 \\ _Charles R Greathouse IV_, Jul 19 2016

%o (Sage) [n^5*(1 + n^4)/2 for n in range(30)] # _G. C. Greubel_, Nov 15 2018

%o (GAP) List([0..30], n -> n^5*(n^4 + 1)/2); # _G. C. Greubel_, Nov 15 2018

%Y Row 9 of A277504.

%Y Cf. A001017 (oriented), A000584 (achiral).

%K nonn,easy

%O 0,3

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