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A168372
a(n) = n^5*(n^4 + 1)/2.
2
0, 1, 272, 9963, 131584, 978125, 5042736, 20185207, 67125248, 193739769, 500050000, 1179054371, 2580014592, 5302435333, 10330792304, 19222059375, 34360262656, 59294648177, 99180589968, 161345086939, 256001600000
OFFSET
0,3
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
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).
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
MATHEMATICA
Table[n^5*(n^4 + 1)/2, {n, 0, 50}] (* G. C. Greubel, Jul 19 2016 *)
PROG
(Magma) [n^5*(n^4+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 28 2011
(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
(GAP) List([0..30], n -> n^5*(n^4 + 1)/2); # G. C. Greubel, Nov 15 2018
CROSSREFS
Row 9 of A277504.
Cf. A001017 (oriented), A000584 (achiral).
Sequence in context: A000517 A350975 A250341 * A023907 A281691 A035842
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 11 2009
STATUS
approved