A170790 a(n) = n^9*(n^8 + 1)/2.

0, 1, 65792, 64579923, 8590065664, 381470703125, 8463334761216, 116315277170407, 1125899973951488, 8338591043543529, 50000000500000000, 252723515428620731, 1109305555950108672, 4325207964992918653

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn
• easy

Comments

Number of unoriented rows of length 17 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)=65792, there are 2^17=131072 oriented arrangements of two colors. Of these, 2^9=512 are achiral. That leaves (131072-512)/2=65280 chiral pairs. Adding achiral and chiral, we get 65792. - 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 (18,-153,816,-3060,8568,-18564, 31824,-43758,48620,-43758,31824,-18564,8568,-3060,816,-153,18,-1).

Crossrefs

Row 17 of A277504.

Cf. A010805 (oriented), A001017 (achiral).

Programs

GAP

List([0..30], n -> n^9*(n^8+1)/2); # G. C. Greubel, Nov 15 2018

Magma

[n^9*(n^8+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

Mathematica

Table[(n^9 (n^8+1))/2,{n,0,20}] (* Harvey P. Dale, Oct 03 2016 *)

PARI

for(n=0,30, print1(n^9*(n^8+1)/2, ", ")) \\ G. C. Greubel, Dec 06 2017

Python

for n in range(0,20): print(int(n**9*(n**8 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

Sage

[n^9*(n^8+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

Formula

G.f.: (x + 65774*x^2 + 63395820*x^3 + 7437692410*x^4 + 236676566180*x^5 + 2858646249342*x^6 + 15527826341908*x^7 + 41568611082650*x^8 + 57445191259830*x^9 + 41568611082650*x^10 + 15527826341908*x^11 + 2858646249342*x^12 + 236676566180*x^13 + 7437692410*x^14 + 63395820*x^15 + 65774*x^16 + x^17)/(1-x)^18. - G. C. Greubel, Dec 06 2017

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

a(n) = (A010805(n) + A001017(n)) / 2 = (n^17 + n^9) / 2.

G.f.: (Sum_{j=1..17} S2(17,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..16} A145882(17,k) * x^k / (1-x)^18.

E.g.f.: (Sum_{k=1..17} S2(17,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>17, a(n) = Sum_{j=1..18} -binomial(j-19,j) * a(n-j). (End)