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%I #12 Jan 08 2016 04:02:31
%S 1,2,8,19,56,147,404,1082,2954,8001,21865,59759,164085,451465,1246358,
%T 3448876,9569376,26611517,74172493,207159274,579724677,1625287220,
%U 4564461309,12839597611,36172421770,102053738981,288317817804,815591326704,2309951078955
%N a(n) = A000081(n+1) - 2*A000081(n).
%C Counts exceptional non-overlapping circles. These circles are exceptional because they are neither generated by encircling any case at level n-1 nor do they result from appending an external circle to any case at level n-1. When n=4 the case is (())(()).
%H Alois P. Heinz, <a href="/A074797/b074797.txt">Table of n, a(n) for n = 4..1000</a>
%e a(8) = 56 because we can write A000081(9) - 2*(A000081(8)= 286 - 2*115 a(8) also = 56 because we know that 8=6+2=5+3=4+4=4+2+2=3+3+2=2+2+2+2 and these partitions contribute 20*1 + 9*2 + 4*5/2 + 4 + 3 + 1 cases.
%p with(numtheory):
%p b:= proc(n) option remember; local d, j; `if` (n<2, n,
%p (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1))
%p end:
%p a:= n-> b(n+1)-2*b(n):
%p seq(a(n), n=4..50); # _Alois P. Heinz_, May 16 2013
%t a81[1] = 1; a81[n_] := a81[n] = Sum[a81[n-k]*DivisorSum[k, #*a81[#]&], {k, 1, n-1}]/(n-1); a[n_] := a81[n+1] - 2*a81[n]; Table[a[n], {n, 4, 50}] (* _Jean-François Alcover_, Jan 08 2016 *)
%K easy,nonn
%O 4,2
%A _Alford Arnold_, Sep 07 2002
%E More terms from _Sascha Kurz_, Feb 10 2003
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