A071236 - OEIS

%I #43 Sep 08 2022 08:45:06

%S 0,1,528,29646,524800,4884375,30236976,141246028,536887296,1743421725,

%T 5000050000,12968792826,30958806528,68929431571,144627596400,

%U 288325575000,549756338176,1007997660153,1785234558096,3065534366950,5120001600000,8339942531151

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

%C Subset of A000217. - _Robert Israel_, Nov 20 2014

%C 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

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

%H Vincenzo Librandi, <a href="/A071236/b071236.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

%F 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

%F From _Robert Israel_, Nov 19 2014: (Start)

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

%F 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)

%F From _Robert A. Russell_, Nov 13 2018: (Start)

%F 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.

%F G.f.: x*Sum_{k=0..9} A145882(10,k) * x^k / (1-x)^11.

%F 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.

%F For n > 10, a(n) = Sum_{j=1..11} -binomial(j-12,j) * a(n-j). (End)

%F 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

%p seq((n^10 + n^5)/2, n=0..100); # _Robert Israel_, Nov 19 2014

%t 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 *)

%o (Magma) [n^5*(n+1)*(n^4-n^3+n^2-n+1)/2: n in [0..40]]; // _Vincenzo Librandi_, Jun 14 2011

%o (PARI) a(n)=binomial(n^5+1,2) \\ _Charles R Greathouse IV_, Nov 19 2014

%o (Sage) [n^5*(1 + n^5)/2 for n in range(40)] # _G. C. Greubel_, Nov 15 2018

%o (GAP) List([0..40], n -> n^5*(1 + n^5)/2); # _G. C. Greubel_, Nov 15 2018

%Y Cf. A000217.

%Y Row 10 of A277504.

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

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Jun 12 2002

%E Definition simplified by _Robert Israel_, Nov 19 2014