A170779 a(n) = n^8*(n^7 + 1)/2.

0, 1, 16512, 7177734, 536903680, 15258984375, 235093332096, 2373783637372, 17592194433024, 102945587570685, 500000050000000, 2088624191887266, 7703511002284032, 25592946914910739, 77784048516800640

Offset

0,3

Comments

Number of unoriented rows of length 15 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)=16512, there are 2^15=32768 oriented arrangements of two colors. Of these, 2^8=256 are achiral. That leaves (32768-256)/2=16256 chiral pairs. Adding achiral and chiral, we get 16512. - 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 (16, -120, 560, -1820, 4368, -8008, 11440, -12870, 11440, -8008, 4368, -1820, 560, -120, 16, -1).

Formula

G.f.: (x + 16496*x^2 + 6913662*x^3 + 424040816*x^4 + 7520608675*x^5 + 51388540128*x^6 + 155693747508*x^7 + 223769408736*x^8 + 155693850903*x^9 + 51388458800*x^10 + 7520620846*x^11 + 424050096*x^12 + 6911077*x^13 + 16256*x^14)/(1-x)^16. - G. C. Greubel, Dec 05 2017

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

a(n) = (A010803(n) + A001016(n)) / 2 = (n^15 + n^8) / 2.

G.f.: (Sum_{j=1..15} S2(15,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..8} S2(8,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

G.f.: x*Sum_{k=0..14} A145882(15,k) * x^k / (1-x)^16.

E.g.f.: (Sum_{k=1..15} S2(15,k)*x^k + Sum_{k=1..8} S2(8,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

For n>15, a(n) = Sum_{j=1..16} -binomial(j-17,j) * a(n-j). (End)

E.g.f.: x*(2 +16510*x +2376067*x^2 +42357651*x^3 +210767970*x^4 + 420693539*x^5 +408741361*x^6 +216627841*x^7 +67128490*x^8 + 12662650*x^9 +1479478*x^10 +106470*x^11 +4550*x^12 +105*x^13 +x^14)*exp(x)/2. - G. C. Greubel, Nov 15 2018

Mathematica

Table[n^8*(n^7+1)/2, {n, 0, 30}] (* G. C. Greubel, Dec 05 2017 *)

Prog

(Magma) [n^8*(n^7+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 26 2011
(PARI) for(n=0, 30, print1(n^8*(n^7+1)/2, ", ")) \\ G. C. Greubel, Dec 05 2017
(Sage) [n^8*(n^7+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

Crossrefs

Row 15 of A277504.

Cf. A010803 (oriented), A001016 (achiral).

Sequence in context: A234047 A223401 A281480 * A251903 A253956 A253963

Adjacent sequences: A170776 A170777 A170778 * A170780 A170781 A170782

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 11 2009

Status

approved