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%I #40 Sep 08 2022 08:45:01
%S 0,0,0,0,0,0,0,60,120,960,1800,9300,16800,71400,126000,480060,834120,
%T 2968560,5103000,17355300,29607600,97567800,165528000,533274060,
%U 901020120,2855012160,4809004200,15050517300,25292030400,78417448200,131542866000,404936532060
%N Number of periodic palindromes using exactly five different symbols.
%D 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 Muniru A Asiru, <a href="/A056491/b056491.txt">Table of n, a(n) for n = 1..600</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,14,-14,-71,71,154,-154,-120, 120).
%F a(n) = 2*A056345(n) - A056285(n).
%F G.f.: -60*x^8*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)). - _Colin Barker_, Jul 08 2012
%F a(n) = (k!/2)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), with k=5 different colors used and where S2(n,k) is the Stirling subset number A008277. - _Robert A. Russell_, Jun 05 2018
%F a(n) = a(n-1) + 14*a(n-2) - 14*a(n-3) - 71*a(n-4) + 71*a(n-5) + 154*a(n-6) - 154*a(n-7) - 120*a(n-8) + 120*a(n-9). - _Muniru A Asiru_, Sep 26 2018
%e For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
%e There are 120 permutations of the five letters used in ABACDEDC. These 120 arrangements can be paired up with a half turn (e.g., ABACDEDC-DEDCABAC) to arrive at the 60 arrangements for n=8. - _Robert A. Russell_, Sep 26 2018
%p with(combinat): a:=n->(factorial(5)/2)*(Stirling2(floor((n+1)/2),5)+Stirling2(ceil((n+1)/2),5)): seq(a(n),n=1..35); # _Muniru A Asiru_, Sep 26 2018
%t k = 5; Table[(k!/2) (StirlingS2[Floor[(n + 1)/2], k] +
%t StirlingS2[Ceiling[(n + 1)/2], k]), {n, 1, 40}] (* _Robert A. Russell_, Jun 05 2018 *)
%t LinearRecurrence[{1, 14, -14, -71, 71, 154, -154, -120, 120}, {0, 0,
%t 0, 0, 0, 0, 0, 60, 120}, 40] (* _Robert A. Russell_, Sep 29 2018 *)
%o (PARI) a(n) = my(k=5); (k!/2)*(stirling(floor((n+1)/2), k, 2) + stirling(ceil((n+1)/2), k, 2)); \\ _Michel Marcus_, Jun 05 2018
%o (GAP) a:=[0,0,0,0,0,0,0,60,120];; for n in [10..35] do a[n]:=a[n-1]+14*a[n-2]-14*a[n-3]-71*a[n-4]+71*a[n-5]+154*a[n-6]-154*a[n-7]-120*a[n-8]+120*a[n-9]; od; a; # _Muniru A Asiru_, Sep 26 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!(-60*x^8*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)))); // _G. C. Greubel_, Oct 13 2018
%Y Cf. A056456.
%Y Column 5 of A305540.
%K nonn,easy
%O 1,8
%A _Marks R. Nester_
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