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A271218 Number of symmetric linked diagrams with n links and no simple link.

%I A271218 #10 Jul 29 2020 05:41:29
%S A271218 1,0,1,3,12,39,167,660,3083,13961,70728,355457,1936449,10587960,
%T A271218 61539129,361182139,2224641540,13880534119,90090083047,593246514588,
%U A271218 4038095508691,27905008440273,198401618299920,1432253086621377,10600146578310209,79639887325700592,611739960145556273
%N A271218 Number of symmetric linked diagrams with n links and no simple link.
%C A271218 Number of symmetric chord diagrams (where reflection is equivalent) with n chords and no simple chords.
%C A271218 Number of symmetric assembly words that do not contain the subword aa.
%D A271218 J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.
%F A271218 a(n) = 2*a(n-1) + (2n-3)*a(n-2) - (2n-5)*a(n-3) + 2*a(n-4) - a(n-5).
%F A271218 a(n) = a(n-1) + 2*(n-1)*a(n-2) + a(n-3) + a(n-4) + 2*sum( k=0..n-4, a(k) ).
%F A271218 a(n) ~ 2^(-1/2) * e^(-5/8) * (2n/e)^(n/2) * e^( sqrt(n/2) )  (conjectured).
%F A271218 a(n)/a(n-1) ~ sqrt(2n)  (conjectured).
%F A271218 a(n)/A047974(n) ~ 1/sqrt(e)  (conjectured).
%e A271218 For n=0 the a(0)=1 solution is { ∅ }.
%e A271218 For n=1 there are no solutions since the link in a diagram with one link, 11, is simple.
%e A271218 For n=2 the a(2)=1 solution is { 1212 }.
%e A271218 For n=3 the a(3)=3 solutions are { 123123, 121323, 123231 }.
%e A271218 For n=4 the a(4)=12 solutions are { 12123434, 12132434, 12324341, 12314234, 12341234, 12342341, 12314324, 12341324, 12343412, 12343421, 12324143, 12342143 }.
%t A271218 RecurrenceTable[{a[n]==2a[n-1]+(2n-3)a[n-2]-(2n-5)a[n-3]+2a[n-4]-a[n-5],a[0]==1,a[1]==0,a[2]==1,a[3]==3,a[4]==12},a[n],{n,20}]
%o A271218 (PARI) lista(nn) = {my(va = vector(nn)); va[1] = 1; va[2] = 0; va[3] = 1; va[4] = 3; va[5] = 12; for (n=5, nn-1, va[n+1] = 2*va[n] + (2*n-3)*va[n-1] - (2*n-5)*va[n-2] + 2*va[n-3] - va[n-4];); va;} \\ _Michel Marcus_, Jul 28 2020
%Y A271218 Cf. A047974, A271215.
%K A271218 nonn,easy
%O A271218 0,4
%A A271218 _Jonathan Burns_, Apr 13 2016
%E A271218 More terms from _Michel Marcus_, Jul 28 2020