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%I #11 Jul 20 2017 01:54:00
%S 1,0,0,1,6,28,127,598,2984,15851,89532,536152,3392609,22609852,
%T 158220300,1159380201,8873605258,70778190768,587125257319,
%U 5055713850058,45114387675316,416535887361323,3973511993495144,39112086371684844
%N a(n) = Bell(n) - Sum_{j=0..n-1} Bell(j), where the Bell numbers are given in A000110.
%C Number of partitions of the set {1,2,...,n} in which n is neither a singleton nor is in a block of consecutive integers. Example: a(4)=6 because we have 14-23, 13-24, 134-2, 124-3, 1-24-3, and 14-2-3. Note that if from the other partitions of {1,2,3,4}, namely 1234, 1-234, 12-34, 1-2-34, 123-4, 1-23-4, 12-3-4, 13-2-4, 1-2-3-4, we delete the blocks containing 4, then we are left with empty, 1, 12, 1-2, 123, 1-23, 12-3, 13-2, 1-2-3, i.e., all the partitions of the sets: empty, {1}, {1,2}, and {1,2,3}.
%C a(n) = A000110(n) - A005001(n).
%F G.f.: G(0)*(1-x-x^2)/(1-x^2) + x/(1-x^2) where G(k) = 1 - x*(1-k*x)/(1 - x - x^2 - (1-2*x-x^2+2*x^3+x^4)/(1 - x - x^2 + (1-k*x)*(k*x+x-1)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Feb 10 2013
%p with(combinat): seq(bell(n)-add(bell(j), j = 0 .. n-1), n = 0 .. 23);
%Y Cf. A000110, A005001.
%K nonn
%O 0,5
%A _Emeric Deutsch_, May 01 2010
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