

A233130


Number of negative formulas with two connectives (> and *) and no variables.


1



1, 0, 2, 6, 38, 210, 1314, 8358, 55118, 370842, 2541626, 17668926, 124321750, 883614498, 6334772562, 45754956054, 332639032734, 2432189656362, 17874332863722, 131957567836206, 978161729926950, 7277592773408562, 54327287358246018, 406792963221032454
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OFFSET

0,3


COMMENTS

More precisely, a(n) is the number of negative formulas containing n connectives * or >, n+1 appearances of symbol "f"=false, and parentheses.
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.
The total number of formulas, both affirmative and negative, with n connectives * or > is A151374(n).


LINKS

Vjekoslav Kovač, Table of n, a(n) for n = 0..1000
V. Čačić and V. Kovač, On the fraction of IL formulas that have normal forms, arXiv:1309.3408 [math.LO], 2013.


FORMULA

Recurrence: a(0)=1, a(n) = sum_{k=0..n1} (A151374(k)a(k)) a(nk1).
G.f.: (3  sqrt(18x) + sqrt(10 + 56x + 6*sqrt(18x))) / 8x.
The ratio a(n)/A151374(n) converges to 1/2  3*sqrt(17)/34 as n>infinity.
Asymptotics: a(n) ~ (1/23*sqrt(17)/34)*8^n/(sqrt(Pi)*n^(3/2)).


EXAMPLE

a(1)=0 because there are no negative formulas with 1 connective.
a(2)=2 because all negative formulas with 2 connectives are: (f>f)>f, (f*f)>f.
a(3)=6 because all negative formulas with 3 connectives are: ((f>f)*f)>f, ((f*f)*f)>f, (f>(f>f))>f, (f>(f*f))>f, (f*(f>f))>f, (f*(f*f))>f.


MATHEMATICA

a[0] = 1;
For[n = 1, n <= 23, n++,
a[n] = Sum[(2^k Binomial[2 k, k]/(k + 1)  a[k]) a[n  k  1], {k,
0, n  1}]];
Table[a[j], {j, 0, 23}]


CROSSREFS

Sequence in context: A202737 A186648 A013033 * A267405 A027322 A085447
Adjacent sequences: A233127 A233128 A233129 * A233131 A233132 A233133


KEYWORD

nonn


AUTHOR

Vjekoslav Kovac, Dec 10 2013


STATUS

approved



