A168664 a(n) = n^7*(n^7 + 1)/2.

0, 1, 8256, 2392578, 134225920, 3051796875, 39182222016, 339111948196, 2199024304128, 11438398618965, 50000005000000, 189874926535206, 641959250190336, 1968688224223903, 5556003465485760, 14596463098125000, 36028797153181696, 84188913484869801

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn
• easy

Comments

Number of unoriented rows of length 14 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)=8256, there are 2^14=16384 oriented arrangements of two colors. Of these, 2^7=128 are achiral. That leaves (16384-128)/2=8128 chiral pairs. Adding achiral and chiral, we get 8256. - 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 (15,-105,455,-1365,3003,-5005,6435,-6435,5005, -3003,1365,-455,105,-15,1).

Crossrefs

Cf. A001015 (Seventh Powers: n^7), A000217 (Triangular Numbers).

Row 14 of A277504.

Cf. A010802 (oriented), A001015 (achiral).

Programs

GAP

List([0..30], n -> n^7*(1 + n^7)/2); # G. C. Greubel, Nov 15 2018

Magma

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

Maple

A168664:=n->n^7*(n^7+1)/2: seq(A168664(n), n=0..20); # Wesley Ivan Hurt, Oct 30 2014

Mathematica

f[n_]:=Module[{c=n^7},c (c+1)/2]; f/@Range[0,30] (* Harvey P. Dale, Mar 19 2011 *)

PARI

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

Sage

[n^7*(1 + n^7)/2 for n in range(40)] # G. C. Greubel, Nov 15 2018

Formula

From Wesley Ivan Hurt, Oct 30 2014: (Start)

G.f.: (x + 8241*x^2 + 2268843*x^3 + 99203675*x^4 + 1285873650*x^5 + 6421633938*x^6 + 13985577438*x^7 + 13985598654*x^8 + 6421628925*x^9 + 1285868525*x^10 + 99207111*x^11 + 2268471*x^12 + 8128*x^13)/(1 - x)^15.

a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15).

a(n) = n^7*(n^7 + 1)/2 = A000217(A001015(n)). (End)

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

a(n) = (A010802(n) + A001015(n)) / 2 = (n^14 + n^7) / 2.

G.f.: (Sum_{j=1..14} S2(14,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..13} A145882(14,k) * x^k / (1-x)^15.

E.g.f.: (Sum_{k=1..14} S2(14,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>14, a(n) = Sum_{j=1..15} -binomial(j-16,j) * a(n-j). (End)

E.g.f.: x*(2+8254*x +789271*x^2 [phone]*x^3 [phone]*x^4 [phone]*x^5 [phone]*x^6 [phone]*x^7 +5135130*x^8 +752752*x^9 + 66066*x^10 +3367*x^11 +91*x^12 +x^13)*exp(x)/2. - G. C. Greubel, Nov 15 2018