A170801 a(n) = n^10*(n^9 + 1)/2.

0, 1, 262656, 581160258, 137439477760, 9536748046875, 304679900238336, 5699447733924196, 72057594574798848, 675425860579888245, 5000000005000000000, 30579545237175985446, 159739999716270145536

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn
• easy

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).

Crossrefs

Row 19 of A277504.

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

Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170796 (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), this sequence (m=9), A170802 (m=10).

Programs

GAP

List([0..30], n -> n^10*(n^9+1)/2); # G. C. Greubel, Nov 15 2018

Magma

[n^10*(n^9+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

Maple

seq(n^10*(n^9 +1)/2, n=0..20); # G. C. Greubel, Oct 11 2019

Mathematica

Table[(n^19 + n^10)/2, {n,0,30}] (* Robert A. Russell, Nov 13 2018 *)

PARI

vector(30, n, n--; n^10*(n^9+1)/2) \\ G. C. Greubel, Nov 15 2018

Sage

[n^10*(n^9+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

Formula

From Robert A. Russell, Nov 13 2018: (Start)

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

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.

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

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.

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