A168663 a(n) = n^7*(n^6 + 1)/2.

0, 1, 4160, 798255, 33562624, 610390625, 6530486976, 48444916975, 274878955520, 1270935305649, 5000005000000, 17261365815551, 53496620605440, 151437584670385, 396857439333824, 973097619609375, 2251799947902976

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn
• easy

Comments

Number of unoriented rows of length 13 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)=4160, there are 2^13=8192 oriented arrangements of two colors. Of these, 2^7=128 are achiral. That leaves (8192-128)/2=4032 chiral pairs. Adding achiral and chiral, we get 4160. - 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 (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Crossrefs

Row 13 of A277504.

Cf. A010801 (oriented), A001015 (achiral).

Programs

Magma

[n^7*(n^6+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

Mathematica

Table[n^7(n^6+1)/2,{n,0,20}] (* Harvey P. Dale, Jan 20 2013 *)

PARI

a(n)=n^7*(n^6+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

Formula

From G. C. Greubel, Jul 28 2016: (Start)

G.f.: x*(1 + 4146*x + 740106*x^2 + 22765250*x^3 + 211641855*x^4 + 752814348*x^5 + 1137578988*x^6 + 752814348*x^7 + 211641855*x^8 + 22765250*x^9 + 740106*x^10 + 4146*x^11 + x^12)/(1 - x)^14.

E.g.f.: (1/2)*x*(2 + 4158*x + 261926*x^2 + 2532880*x^3 + 7508641*x^4 + 9321333*x^5 + 5715425*x^6 + 1899612*x^7 + 359502*x^8 + 39325*x^9 + 2431*x^10 + 78*x^11 + x^12)*exp(x). (End)

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

a(n) = (A010801(n) + A001015(n)) / 2 = (n^13 + n^7) / 2.

G.f.: (Sum_{j=1..13} S2(13,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

G.f.: x*Sum_{k=0..12} A145882(13,k) * x^k / (1-x)^14.

E.g.f.: (Sum_{k=1..13} S2(13,k)*x^k + Sum_{k=1..7} S2(7,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

For n>13, a(n) = Sum_{j=1..14} -binomial(j-15,j) * a(n-j). (End)