This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A056312 #37 Sep 08 2022 08:45:01
%S A056312 0,0,0,0,60,900,8400,63000,417120,2551560,14804700,82764900,450518460,
%T A056312 2404510500,12646078200,65771496000,339165516120,1737486149760,
%U A056312 8855359634100,44952367981500,227475768907860,1148269329527100,5785013373810000,29100047092479000
%N A056312 Number of reversible strings with n beads using exactly five different colors.
%C A056312 A string and its reverse are considered to be equivalent.
%D A056312 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
%H A056312 Vincenzo Librandi, <a href="/A056312/b056312.txt">Table of n, a(n) for n = 1..1000</a>
%H A056312 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (13,-45,-75,695,-575,-3195,5595,4706,-14918,2160,12840,-7200).
%F A056312 a(n) = A032122(n) - 5*A032121(n) + 10*A032120(n) - 10*A005418(n+1) + 5.
%F A056312 G.f.: -60*x^5*(120*x^7 - 17*x^6 - 50*x^5 - 32*x^4 + 20*x^3 + 10*x^2 - 2*x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)). [_Colin Barker_, Sep 03 2012]
%F A056312 a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=5 is the number of colors and S2 is the Stirling subset number. - _Robert A. Russell_, Sep 25 2018
%e A056312 For n=5, the 60 rows are 60 permutations of ABCDE that do not include any mutual reversals. Each of the 60 chiral pairs, such as ABCDE-EDCBA, is then counted just once.
%t A056312 k=5; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,30}] (* _Robert A. Russell_, Nov 25 2017 *) adapted
%t A056312 CoefficientList[Series[-60*x^4*(120*x^7 - 17*x^6 - 50*x^5 - 32*x^4 + 20*x^3 + 10*x^2 - 2*x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)), {x, 0, 30}], x] (* _Stefano Spezia_, Sep 29 2018 *)
%o A056312 (PARI) a(n) = 60*(stirling(n, 5, 2) + stirling(ceil(n/2), 5, 2)); \\ _Altug Alkan_, Sep 27 2018
%o A056312 (Magma) [60*(StirlingSecond(n, 5)+StirlingSecond(Ceiling(n/2), 5)): n in [1..30]]; // _Vincenzo Librandi_, Sep 30 2018
%Y A056312 Cf. A005418, A032120, A032121, A032122.
%Y A056312 Column 5 of A305621.
%Y A056312 Equals (A001118 + A056456) / 2 = A001118 - A305625 = A305625 + A056456.
%K A056312 nonn,easy
%O A056312 1,5
%A A056312 _Marks R. Nester_