A170801 - OEIS

%I #24 Sep 08 2022 08:45:49

%S 0,1,262656,581160258,137439477760,9536748046875,304679900238336,

%T 5699447733924196,72057594574798848,675425860579888245,

%U 5000000005000000000,30579545237175985446,159739999716270145536

%N a(n) = n^10*(n^9 + 1)/2.

%C Number of unoriented rows of length 19 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)=262656, there are 2^19=524288 oriented arrangements of two colors. Of these, 2^10=1024 are achiral. That leaves (524288-1024)/2=261632 chiral pairs. Adding achiral and chiral, we get 262656. - _Robert A. Russell_, Nov 13 2018

%H Vincenzo Librandi, <a href="/A170801/b170801.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).

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

%F a(n) = (A010807(n) + A008454(n)) / 2 = (n^19 + n^10) / 2.

%F G.f.: (Sum_{j=1..19} S2(19,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..10} S2(10,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

%F G.f.: x*Sum_{k=0..18} A145882(19,k) * x^k / (1-x)^20.

%F E.g.f.: (Sum_{k=1..19} S2(19,k)*x^k + Sum_{k=1..10} S2(10,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

%F For n>19, a(n) = Sum_{j=1..20} -binomial(j-21,j) * a(n-j). (End)

%p seq(n^10*(n^9 +1)/2, n=0..20); # _G. C. Greubel_, Oct 11 2019

%t Table[(n^19 + n^10)/2, {n,0,30}] (* _Robert A. Russell_, Nov 13 2018 *)

%o (Magma)[n^10*(n^9+1)/2: n in [0..20]]; // _Vincenzo Librandi_, Aug 27 2011

%o (PARI) vector(30, n, n--; n^10*(n^9+1)/2) \\ _G. C. Greubel_, Nov 15 2018

%o (Sage) [n^10*(n^9+1)/2 for n in range(30)] # _G. C. Greubel_, Nov 15 2018

%o (GAP) List([0..30], n -> n^10*(n^9+1)/2); # _G. C. Greubel_, Nov 15 2018

%Y Row 19 of A277504.

%Y Cf. A010807 (oriented), A008454 (achiral).

%Y Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170896 (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), this sequence (m=9), A170802 (m=10).

%K nonn,easy

%O 0,3

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