

A005612


Number of Boolean functions of n variables that are variously called "unate cascades" or "1decision list functions" or "readonce threshold functions".
(Formerly M1895)


6



2, 8, 64, 736, 10624, 183936, 3715072, 85755392, 2226939904, 64255903744, 2039436820480, 70614849282048, 2648768014680064, 106998205418995712, 4630973410260287488, 213794635951073787904, 10486975675879356104704
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OFFSET

1,1


COMMENTS

Several other characterizations are given in the paper by Eitel et al.
These functions are the Boolean functions with the nice property that all of their projections are "canalizing" or "singlefaced": that is, f is constant on half of the ncube and on the other half it recursively satisfies the same constraint.


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Herman Jamke and Robert Israel, Table of n, a(n) for n = 1..350 (n = 1..22 from Herman Jamke)
E. A. Bender and J. T. Butler, Asymptotic approximations for the number of fanoutfree functions, IEEE Trans. Computers, 27 (1978), 11801183. (Annotated scanned copy)
J. T. Butler, Letter to N. J. A. Sloane, Dec. 1978.
Thomas Eiter, Toshihide Ibaraki and Kazuhisa Makino, Decision lists and related Boolean functions, Theoretical Computer Science 270 (2002), 493524.
A. S. Jarraha, B. Raposab and R. Laubenbachera, Nested canalyzing, unate cascade and polynomial functions, Physica D: Nonlinear Phenomena Volume 233, Issue 2, 15 September 2007, 167174.
T. Sasao, K. Kinoshita, On the Number of FanoutFree Functions and Unate Cascade Functions, IEEE Transactions on Computers, Volume C28, Issue 1 (1979), 6672.
Index entries for sequences related to Boolean functions


FORMULA

When n > 1, the number is 2^{n+1}(P_nnP_{n1}), where P_n is the number of weak orders (preferential arrangements), sequence A000670. For example, when n=4 we have 736 = 32 times (75  4*13).
Bender and Butler give the e.g.f. 2(x+e^{2x}1)/(12e^{2x}), which can easily be simplified to (24x)/(2e^(2x))+2x2.
a(n) ~ n! * (1  log(2)) * 2^n / (log(2))^(n+1).  Vaclav Kotesovec, Nov 27 2017


MAPLE

egf:= (24*x)/(2exp(2*x))+2*x2:
S:=series(egf, x, 31):
seq(j! *coeff(S, x, j), j=1..30); # Robert Israel, Jul 07 2015


MATHEMATICA

p[0] = 1; p[n_] := p[n] = Sum[Binomial[n, k]*p[nk], {k, 1, n}]; a[n_] := a[n] = 2^(n+1)*(p[n]  n*p[n1]); a[1] = 2; Table[a[n], {n, 1, 17}] (* JeanFrançois Alcover, Aug 01 2011, after formula *)


PROG

(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst((24*y)/(2exp(2*y))+2*y2, y, x+x*O(x^n)), n)) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008


CROSSREFS

See also sequence A005840, which is A005612 divided by 2^n. These are the monotone functions of the kind enumerated in the present sequence.
Sequence in context: A059862 A268666 A193549 * A136282 A092934 A224801
Adjacent sequences: A005609 A005610 A005611 * A005613 A005614 A005615


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Better description, comments, formulas and a new reference from Don Knuth, Sep 22 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008


STATUS

approved



