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%I M1285 N0492 #92 Oct 30 2023 13:26:26
%S 2,4,14,104,1882,94572,15028134,8378070864,17561539552946,
%T 144130531453121108
%N Number of threshold functions of n or fewer variables.
%C a(n) is also equal to the number of self-dual threshold functions of n+1 or fewer variables. - Alastair D. King, Mar 17, 2023.
%D Sze-Tsen Hu, Threshold Logic, University of California Press, 1965 see page 57.
%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
%D S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 3.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D C. Stenson, Weighted voting, threshold functions, and zonotopes, in The Mathematics of Decisions, Elections, and Games, Volume 625 of Contemporary Mathematics Editors Karl-Dieter Crisman, Michael A. Jones, American Mathematical Society, 2014, ISBN 0821898663, 9780821898666
%H Taylor Brysiewicz, Holger Eble, and Lukas Kühne, <a href="https://arxiv.org/abs/2105.14542">Enumerating chambers of hyperplane arrangements with symmetry</a>, arXiv:2105.14542 [math.CO], 2021.
%H Nicolle Gruzling, <a href="https://doi.org/10.24124/2007/bpgub464">Linear separability of the vertices of an n-dimensional hypercube</a>, M.Sc Thesis, University of Northern British Columbia, 2006. [From W. Lan (wl(AT)fjrtvu.edu.cn), Jun 27 2010]
%H Samuel C. Gutekunst, Karola Mészáros, and T. Kyle Petersen, <a href="https://arxiv.org/abs/1903.06595">Root Cones and the Resonance Arrangement</a>, arXiv:1903.06595 [math.CO], 2019.
%H Alastair D. King, <a href="/A002080/a002080.pdf">Comments on A002080 and related sequences based on threshold functions</a>, Mar 17 2023.
%H Isaac K. Martin, Andrew G. Moore, John T. Daly, Jess J. Meyer, and Teresa M. Ranadive, <a href="https://arxiv.org/abs/2310.16246">Design of General Purpose Minimal-Auxiliary Ising Machines</a>, arXiv:2310.16246 [math.OC], 2023. See p. 7.
%H Chris Mingard, Joar Skalse, Guillermo Valle-Pérez, David Martínez-Rubio, Vladimir Mikulik, and Ard A. Louis, <a href="https://arxiv.org/abs/1909.11522">Neural networks are a priori biased towards Boolean functions with low entropy</a>, arXiv:1909.11522 [cs.LG], 2019.
%H Guido F. Montufar and Jason Morton, <a href="https://arxiv.org/abs/1206.0387">When Does a Mixture of Products Contain a Product of Mixtures?</a>, arXiv preprint arXiv:1206.0387 [stat.ML], 2012-2014.
%H S. Muroga, <a href="/A000371/a000371.pdf">Threshold Logic and Its Applications</a>, Wiley, NY, 1971 [Annotated scans of a few pages]
%H Muroga, Saburo, Iwao Toda, and Satoru Takasu, <a href="/A002079/a002079.pdf">Theory of majority decision elements</a>, Journal of the Franklin Institute 271.5 (1961): 376-418. [Annotated scans of pages 413 and 414 only]
%H S. Muroga, T. Tsuboi and C. R. Baugh, <a href="https://people.eecs.berkeley.edu/~alanmi/publications/other/01671639_002.pdf">Enumeration of threshold functions of eight variables</a>, IEEE Trans. Computers, 19 (1970), 818-825.
%H S. Muroga, T. Tsuboi and C. R. Baugh, <a href="/A002077/a002077.pdf">Enumeration of threshold functions of eight variables</a>, IEEE Trans. Computers, 19 (1970), 818-825. [Annotated scanned copy]
%H Michael Z. Spivey and Laura L. Steil, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H Stephen Wolfram, A New Kind Of Science. <a href="http://www.wolframscience.com/nksonline/page-1102a-text">page 1102</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Linear_separability">Linear separability</a> [From W. Lan (wl(AT)fjrtvu.edu.cn), Jun 27 2010]
%H R. O. Winder, <a href="https://doi.org/10.1109/PGEC.1965.264136">Enumeration of seven-argument threshold functions</a>, IEEE Trans. Electron. Computers, 14 (1965), 315-325.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>
%F a(n) = Sum_{k=0..n} A000615(k)*binomial(n,k) = Sum_{k=0..n} A002079(k)*binomial(n,k)*2^k. Also A002078(n) = (1/2^n)*Sum_{k=0..n} a(k)*binomial(n,k), a(n-1) = Sum_{k=1..n} A002077(k)*binomial(n,k)*2^k, and A002080(n) = (1/2^n)*Sum_{k=1..n} a(k)*binomial(n,k). - Alastair D. King, Mar 17, 2023.
%Y Cf. A000615, A002077-A002080, A109456, A116986.
%K nonn,hard,core,nice,more
%O 0,1
%A _N. J. A. Sloane_
%E a(9) from _Minfeng Wang_, Jun 27 2010
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