A074797 a(n) = A000081(n+1) - 2*A000081(n).

1, 2, 8, 19, 56, 147, 404, 1082, 2954, 8001, 21865, 59759, 164085, 451465, 1246358, 3448876, 9569376, 26611517, 74172493, 207159274, 579724677, 1625287220, 4564461309, 12839597611, 36172421770, 102053738981, 288317817804, 815591326704, 2309951078955

Offset: 4

Author

Alford Arnold, Sep 07 2002

Keywords

• easy
• nonn

Comments

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 (())(()).

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..1000

Crossrefs

Cf. A000081.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<2, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1))
    end:
a:= n-> b(n+1)-2*b(n):
seq(a(n), n=4..50);  # Alois P. Heinz, May 16 2013

Mathematica

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 *)

Extensions

More terms from Sascha Kurz, Feb 10 2003