A168194 a(n) = n^4*(n^3 + 1)/2.

0, 1, 72, 1134, 8320, 39375, 140616, 412972, 1050624, 2394765, 5005000, 9750906, 17926272, 31388539, 52725960, 85455000, 134250496, 205211097, 306162504, 447001030, 640080000, 900641511, 1247296072, 1702552644, 2293401600

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn
• easy

Comments

Number of unoriented rows of length 7 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)=72, there are 2^7=128 oriented arrangements of two colors. Of these, 2^4=16 are achiral. That leaves (128-16)/2=56 chiral pairs. Adding achiral and chiral, we get 72. - Robert A. Russell, Nov 13 2018

Links

• Vincenzo Librandi and Bruno Berselli, Table of n, a(n) for n = 0..1000 (first 480 terms from Vincenzo Librandi).
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Row 7 of A277504.

Cf. A001015 (oriented), A000583 (achiral).

Programs

Magma

[n^4*(n^3+1)/2: n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Mathematica

Table[(n^4 (n^3+1))/2,{n,0,40}] (* Harvey P. Dale, Apr 29 2011 *)

PARI

vector(50, n, n--; n^4*(n^3+1)/2) \\ G. C. Greubel, Nov 14 2018

Sage

[n^4*(n^3+1)/2 for n in (0..50)] # G. C. Greubel, Nov 14 2018

Formula

G.f.: x*(1 + 64*x + 586*x^2 + 1208*x^3 + 605*x^4 + 56*x^5)/(1-x)^8. - Colin Barker, Apr 26 2012

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

a(n) = (A001015(n) + A000583(n)) / 2 = (n^7 + n^4) / 2.

G.f.: (Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..4} S2(4,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

G.f.: x*Sum_{k=0..6} A145882(7,k) * x^k / (1-x)^8.

E.g.f.: (Sum_{k=1..7} S2(7,k)*x^k + Sum_{k=1..4} S2(4,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

For n>7, a(n) = Sum_{j=1..8} -binomial(j-9,j) * a(n-j). (End)

E.g.f.: x*(2 +70*x +307*x^2 +351*x^3 +140*x^4 +21*x^5 +x^6)*exp(x)/2. - G. C. Greubel, Nov 14 2018