A168627 a(n) = n^6*(n^5 + 1)/2.

0, 1, 1056, 88938, 2099200, 24421875, 181421856, 988722196, 4295098368, 15690795525, 50000500000, 142656721086, 371505678336, 896082610423, 2024786349600, 4324883625000, 8796101410816, 17135960222601

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn

Comments

Number of unoriented rows of length 11 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)=1056, there are 2^11=2048 oriented arrangements of two colors. Of these, 2^6=64 are achiral. That leaves (2048-64)/2=992 chiral pairs. Adding achiral and chiral, we get 1056. - Robert A. Russell, Nov 13 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1).

Crossrefs

Row 11 of A277504.

Cf. A008455 (oriented), A001014 (achiral).

Programs

GAP

List([0..30], n -> n^6*(1 + n^5)/2); # G. C. Greubel, Nov 15 2018

Magma

[n^6*(1 + n^5)/2: n in [0..30]]; // G. C. Greubel, Nov 15 2018

Mathematica

Table[n^6*(n^5+1)/2, {n, 0, 30}] (* G. C. Greubel, Jul 27 2016 *)
LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{0,1,1056,88938,2099200,24421875,181421856,988722196,4295098368,15690795525,50000500000,142656721086},20] (* Harvey P. Dale, Nov 21 2024 *)

PARI

vector(30, n, n--; n^6*(1 + n^5)/2) \\ G. C. Greubel, Nov 15 2018

Sage

[n^6*(1 + n^5)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

Formula

From G. C. Greubel, Jul 27 2016: (Start)

G.f.: x*(1 + 1044*x + 76332*x^2 + 1101420*x^3 + 4869558*x^4 + 7862124*x^5 + 4868556*x^6 + 1102068*x^7 + 76305*x^8 + 992*x^9)/(1 - x)^12.

E.g.f.: (1/2)* x *(2 + 1054*x + 28591*x^2 + 145815*x^3 + 246745*x^4 + 179488*x^5 + 63987*x^6 + 11880*x^7 + 1155*x^8 + 55*x^9 + x^10)*exp(x). (End)

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

a(n) = (A008455(n) + A001014(n)) / 2 = (n^11 + n^6) / 2.

G.f.: (Sum_{j=1..11} S2(11,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..6} S2(6,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

G.f.: x*Sum_{k=0..10} A145882(11,k) * x^k / (1-x)^12.

E.g.f.: (Sum_{k=1..11} S2(11,k)*x^k + Sum_{k=1..6} S2(6,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

For n>11, a(n) = Sum_{j=1..12} -binomial(j-13,j) * a(n-j). (End)