A170802 - OEIS

%I #32 Sep 08 2022 08:45:49

%S 0,1,524800,1743421725,549756338176,47683720703125,1828079250264576,

%T 39896133290043625,576460752840294400,6078832731271856601,

%U 50000000005000000000,336374997479248716901,1916879996254696243200

%N a(n) = n^10*(n^10 + 1)/2.

%C By definition, all terms are triangular numbers. - _Harvey P. Dale_, Aug 12 2012

%C Number of unoriented rows of length 20 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)=524800, there are 2^20=1048576 oriented arrangements of two colors. Of these, 2^10=1024 are achiral. That leaves (1048576-1024)/2=523776 chiral pairs. Adding achiral and chiral, we get 524800. - _Robert A. Russell_, Nov 13 2018

%H Vincenzo Librandi, <a href="/A170802/b170802.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

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

%F a(n) = (A010808(n) + A008454(n)) / 2 = (n^20 + n^10) / 2.

%F G.f.: (Sum_{j=1..20} S2(20,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..10} S2(10,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

%F G.f.: x*Sum_{k=0..19} A145882(20,k) * x^k / (1-x)^21.

%F E.g.f.: (Sum_{k=1..20} S2(20,k)*x^k + Sum_{k=1..10} S2(10,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

%F For n>20, a(n) = Sum_{j=1..21} -binomial(j-22,j) * a(n-j). (End)

%p seq(n^10*(n^10 +1)/2, n=0..20); # _G. C. Greubel_, Oct 11 2019

%t n10[n_]:=Module[{c=n^10},(c(c+1))/2];Array[n10,15,0] (* _Harvey P. Dale_, Jul 17 2012 *)

%o (Magma) [n^10*(n^10+1)/2: n in [0..20]]; // _Vincenzo Librandi_, Aug 27 2011

%o (PARI) vector(30, n, n--; n^10*(n^10+1)/2) \\ _G. C. Greubel_, Nov 15 2018

%o (Sage) [n^10*(n^10+1)/2 for n in range(30)] # _G. C. Greubel_, Nov 15 2018

%o (GAP) List([0..30], n -> n^10*(n^10+1)/2); # _G. C. Greubel_, Nov 15 2018

%o (Python) for n in range(0,20): print(int(n**10*(n**10 + 1)/2), end=', ') # _Stefano Spezia_, Nov 15 2018

%Y Row 20 of A277504.

%Y Cf. A010808 (oriented), A008454 (achiral).

%Y Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170896 (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), A170801 (m=9), this sequence (m=10).

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Dec 11 2009