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%I #31 Nov 09 2022 10:01:41
%S 0,2,6,34,186,1134,7134,46554,310962,2118502,14657478,102724802,
%T 727695402,5202222302,37483252398,271925724906,1984561228386,
%U 14560612258518,107335786509078,794597315522514,5904817403594970,44031888040118478,329378395247260542,2470999656320266746
%N Number of affirmative formulas with two connectives (-> and *) and no variables.
%C a(n) is the number of affirmative formulas containing n connectives * or ->, n+1 appearances of symbol "f"=false, and parentheses.
%C Each negative formula is either "f", or is of the form "A->N", where A is a simpler affirmative formula and N is a simpler negative formula. Affirmative formulas are precisely those that are not negative.
%C The total number of formulas, both affirmative and negative, with n connectives * or -> is A151374(n).
%H Vjekoslav Kovač, <a href="/A233396/b233396.txt">Table of n, a(n) for n = 0..1000</a>
%H V. Čačić, V. Kovač, <a href="http://arxiv.org/abs/1309.3408">On the fraction of IL formulas that have normal forms</a>, arXiv:1309.3408 [math.LO], 2013.
%F a(0)=0, a(n) = A151374(n) - Sum_{k=0..n-1} (A151374(k) - a(k))*a(n-k-1).
%F G.f.: (5 - sqrt(1-8*x) - sqrt(10 + 56*x + 6*sqrt(1-8*x))) / (8*x). [Typo corrected by _Georg Fischer_, Nov 09 2022]
%F The ratio a(n)/A151374(n) converges to 1/2 + 3*sqrt(17)/34 as n->infinity.
%F Asymptotics: a(n) ~ (1/2 + 3*sqrt(17)/34)*8^n/(sqrt(Pi)*n^(3/2)).
%e a(1)=2 because all affirmative formulas with 1 connective are: f->f, f*f.
%e a(2)=6 because all affirmative formulas with 2 connectives are: (f->f)*f, (f*f)*f, f->(f->f), f->(f*f), f*(f->f), f*(f*f).
%t a[0] = 0;
%t For[n = 1, n <= 23, n++,
%t a[n] = 2^n Binomial[2 n, n]/(n + 1) -
%t Sum[(2^k Binomial[2 k, k]/(k + 1) - a[k]) a[n - k - 1], {k, 0,
%t n - 1}]];
%t Table[a[j], {j, 0, 23}] (* or *)
%t CoefficientList[Series[(5 - Sqrt[1-8x] - Sqrt[10 + 56x + 6*Sqrt[1-8x]]) / (8*x),{x,0,35}], x] (* _Georg Fischer_, Nov 09 2022 *)
%K nonn
%O 0,2
%A _Vjekoslav Kovac_, Dec 10 2013
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