A168372 a(n) = n^5*(n^4 + 1)/2.

0, 1, 272, 9963, 131584, 978125, 5042736, 20185207, 67125248, 193739769, 500050000, 1179054371, 2580014592, 5302435333, 10330792304, 19222059375, 34360262656, 59294648177, 99180589968, 161345086939, 256001600000

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn
• easy

Comments

Number of unoriented rows of length 9 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)=272, there are 2^9=512 oriented arrangements of two colors. Of these, 2^5=32 are achiral. That leaves (512-32)/2=240 chiral pairs. Adding achiral and chiral, we get 272. - 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 (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Crossrefs

Row 9 of A277504.

Cf. A001017 (oriented), A000584 (achiral).

Programs

GAP

List([0..30], n -> n^5*(n^4 + 1)/2); # G. C. Greubel, Nov 15 2018

Magma

[n^5*(n^4+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 28 2011

Mathematica

Table[n^5*(n^4 + 1)/2, {n,0,50}] (* G. C. Greubel, Jul 19 2016 *)

PARI

a(n)=n^5*(n^4+1)/2 \\ Charles R Greathouse IV, Jul 19 2016

Sage

[n^5*(1 + n^4)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

Formula

G.f.: x*(1 + 262*x + 7288*x^2 + 44074*x^3 + 78190*x^4 + 44074*x^5 + 7288*x^6 + 262*x^7 + x^8)/(1 - x)^10. - G. C. Greubel, Jul 19 2016

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

a(n) = (A001017(n) + A000584(n)) / 2 = (n^9 + n^5) / 2.

G.f.: (Sum_{j=1..9} S2(9,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..5} S2(5,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

G.f.: x*Sum_{k=0..8} A145882(9,k) * x^k / (1-x)^10.

E.g.f.: (Sum_{k=1..9} S2(9,k)*x^k + Sum_{k=1..5} S2(5,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

For n>9, a(n) = Sum_{j=1..10} -binomial(j-11,j) * a(n-j). (End)

E.g.f.: x*(2 +270*x +3050*x^2 +7780*x^3 +6952*x^4 +2646*x^5 +462*x^6 + 36*x^7 +x^8)*exp(x)/2. - G. C. Greubel, Nov 15 2018