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%I #20 Sep 08 2022 08:46:01
%S 1,71,918,6667,33665,131616,425866,1192178,2977857,6785605,14339006,
%T 28451061,53519713,96176822,166119570,277155796,448497281,706337523,
%U 1085753062,1632969935,2408039361,3487969276,4970360858,6977601702,9661669825,13209605201,17849708046
%N Number of bilaterally asymmetric 8-hoops with n symbols and no a-rooted trees.
%C Follows from the polynomial of eq (29) in the Williamson paper and differs from A210768 (the published version) in a(3) and a(5).
%H Vincenzo Librandi, <a href="/A210699/b210699.txt">Table of n, a(n) for n = 2..1000</a>
%H Williamson, S. G. <a href="http://dx.doi.org/10.1016/0012-365X(72)90043-X">The combinatorial analysis of patterns and the principle of inclusion-exclusion</a>. Discrete Math. 1 (1972), no. 4, 357--388. MR0299493 (45 #8541)
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = (n-1)*(n^7-7*n^6+29*n^5-71*n^4+116*n^3-128*n^2+80*n-32)/16.
%F G.f.: x^2*(1+62*x+315*x^2+877*x^3+872*x^4+351*x^5+40*x^6+2*x^7)/(1-x)^9. [_Colin Barker_, Apr 01 2012]
%F a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). _Vincenzo Librandi_, May 13 2012
%p A210768 := proc(n)
%p (n^8 -8*n^7 +36*n^6 -100*n^5 +187*n^4 -244*n^3 +208*n^2 -112*n+32)/16 ;
%p end proc:
%p seq(A210768(n),n=2..20) ;
%t CoefficientList[Series[(1+62*x+315*x^2+877*x^3+872*x^4+351*x^5+ 40*x^6+ 2*x^7)/(1-x)^9,{x,0,30}],x] (* _Vincenzo Librandi_, May 13 2012 *)
%o (Magma) I:=[1, 71, 918, 6667, 33665, 131616, 425866, 1192178, 2977857]; [n le 9 select I[n] else 9*Self(n-1)-36*Self(n-2)+84*Self(n-3)-126*Self(n-4)+126*Self(n-5)-84*Self(n-6)+36*Self(n-7)-9*Self(n-8)+Self(n-9): n in [1..30]]; // _Vincenzo Librandi_, May 13 2012
%K nonn,easy
%O 2,2
%A _R. J. Mathar_, Mar 30 2012
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