A170791 a(n) = n^9*(n^9 + 1)/2.

0, 1, 131328, 193720086, 34359869440, 1907349609375, 50779983373056, 814206819132028, 9007199321849856, 75047317842209805, 500000000500000000, 2779958657925089586, 13311666643022512128, 56227703481280946251

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn
• easy

Comments

Number of unoriented rows of length 18 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)=131328, there are 2^18=262144 oriented arrangements of two colors. Of these, 2^9=512 are achiral. That leaves (262144-512)/2=130816 chiral pairs. Adding achiral and chiral, we get 131328. - 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 (19,-171,969,-3876,11628,-27132, 50388,-75582,92378,-92378,75582,-50388,27132,-11628,3876,-969,171, -19,1).

Crossrefs

Row 18 of A277504.

Cf. A010806 (oriented), A001017 (achiral).

Programs

GAP

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

Magma

[n^9*(n^9+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

Mathematica

f[n_]:=Module[{n9=n^9},(n9(n9+1))/2]; Array[f,20,0] (* Harvey P. Dale, Nov 24 2012 *)
Table[n^9*(n^9+1)/2, {n,0,30}] (* G. C. Greubel, Dec 06 2017 *)

PARI

for(n=0,30, print1(n^9*(n^9+1)/2, ", ")) \\ G. C. Greubel, Dec 06 2017

Python

for n in range(0,20): print(int(n**9*(n**9 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

Sage

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

Formula

G.f.: (x + 131309*x^2 + 191225025*x^3 + 30701643925*x^4 + 1287510971765*x^5 + 20228672721537*x^6 + 142998536758213*x^7 + 503354983579865*x^8 + 932692830330915*x^9 + 932692827449735*x^10 + 503354984335363*x^11 + 142998537549087*x^12 + 20228672026535*x^13 + 1287511125835*x^14 + 30701669175*x^15 + 191214899*x^16 + 130816*x^17) /(1-x)^19. - G. C. Greubel, Dec 06 2017

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

a(n) = (A010806(n) + A001017(n)) / 2 = (n^18 + n^9) / 2.

G.f.: (Sum_{j=1..18} S2(18,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..9} S2(9,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

G.f.: x*Sum_{k=0..17} A145882(18,k) * x^k / (1-x)^19.

E.g.f.: (Sum_{k=1..18} S2(18,k)*x^k + Sum_{k=1..9} S2(9,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

For n>18, a(n) = Sum_{j=1..19} -binomial(j-20,j) * a(n-j). (End)