A071236 a(n) = (n^10 + n^5)/2.

0, 1, 528, 29646, 524800, 4884375, 30236976, 141246028, 536887296, 1743421725, 5000050000, 12968792826, 30958806528, 68929431571, 144627596400, 288325575000, 549756338176, 1007997660153, 1785234558096, 3065534366950, 5120001600000, 8339942531151

Offset

0,3

Comments

Subset of A000217. - Robert Israel, Nov 20 2014

Number of unoriented rows of length 10 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)=528, there are 2^10=1024 oriented arrangements of two colors. Of these, 2^5=32 are achiral. That leaves (1024-32)/2=496 chiral pairs. Adding achiral and chiral, we get 528. - Robert A. Russell, Nov 13 2018

References

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Formula

a(0)=0, a(1)=1, a(2)=528, a(3)=29646, a(4)=524800, a(5)=4884375, a(6)=30236976, a(7)=141246028, a(8)=536887296, a(9)=1743421725, a(10)=5000050000, a(n)= 11*a(n-1)- 55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+ 55*a(n-9)- 11*a(n-10)+a(n-11). - Harvey P. Dale, Jul 24 2012

From Robert Israel, Nov 19 2014: (Start)

a(n) = (A008454(n) + A000584(n))/2 = n^5*(n+1)*(n^4 -n^3 +n^2 -n +1)/2.

G.f.: x*(1 +517*x +23893*x^2 +227569*x^3 +655315*x^4 +655039*x^5 +227623*x^6 +23947*x^7 +496*x^8)/(1-x)^11. (End)

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

G.f.: (Sum_{j=1..10} S2(10,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..9} A145882(10,k) * x^k / (1-x)^11.

E.g.f.: (Sum_{k=1..10} S2(10,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 > 10, a(n) = Sum_{j=1..11} -binomial(j-12,j) * a(n-j). (End)

E.g.f.: x*(2 + 526*x + 9355*x^2 + 34115 x^3 + 42526*x^4 + 22827*x^5 + 5880*x^6 + 750*x^7 + 45*x^8 + x^9)*exp(x)/2. - G. C. Greubel, Nov 15 2018

Maple

seq((n^10 + n^5)/2, n=0..100); # Robert Israel, Nov 19 2014

Mathematica

Table[n^5(n+1)(n^4-n^3+n^2-n+1)/2, {n, 0, 30}] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {0, 1, 528, 29646, 524800, 4884375, 30236976, 141246028, 536887296, 1743421725, 5000050000}, 30](* Harvey P. Dale, Jul 24 2012 *)

Prog

(Magma) [n^5*(n+1)*(n^4-n^3+n^2-n+1)/2: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011
(PARI) a(n)=binomial(n^5+1, 2) \\ Charles R Greathouse IV, Nov 19 2014
(Sage) [n^5*(1 + n^5)/2 for n in range(40)] # G. C. Greubel, Nov 15 2018
(GAP) List([0..40], n -> n^5*(1 + n^5)/2); # G. C. Greubel, Nov 15 2018

Crossrefs

Cf. A000217.

Row 10 of A277504.

Cf. A008454 (oriented), A000584 (achiral).

Sequence in context: A233103 A223244 A059465 * A048101 A233087 A008690

Adjacent sequences: A071233 A071234 A071235 * A071237 A071238 A071239

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 12 2002

Extensions

Definition simplified by Robert Israel, Nov 19 2014

Status

approved