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A016269
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Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.
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63
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1, 9, 55, 285, 1351, 6069, 26335, 111645, 465751, 1921029, 7859215, 31964205, 129442951, 522538389, 2104469695, 8460859965, 33972448951, 136276954149, 546269553775, 2188563950925, 8764714059751
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Half the number of 2 X (n+2) binary arrays with both a path of adjacent 1's and a path of adjacent 0's from top row to bottom row. - R. H. Hardin (rhhardin(AT)att.net), Mar 21 2002
As (0,0,1,9,55,...) this is the third binomial transform of cosh(x)-1. It is the binomial transform of A000392, when this has two leading zeros. Its e.g.f. is then exp(3x)cosh(x)-exp(3x) and a(n)=(4^n-2*3^n+2^n)/2. - Paul Barry (pbarry(AT)wit.ie), May 13 2003
Let P(A) be the power set of an n-element set A. Then a(n-2) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008
a(n) gives also the third column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. [From Wolfdieter Lang, Oct 08 2011]
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,2).
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (9,-26,24).
K. S. Brown, Dedekind's problem
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Index entries for sequences related to Boolean functions
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
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FORMULA
| G.f.: 1/((1-2x)(1-3x)(1-4x)). a(n) = (2^n)*(2^n - 1)/2 - 3^n + 2^n.
a(n)=sum{0<=i,j,k,<=n, i+j+k=n, 2^i*3^j*4^k}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
a(n)=2^(n+1)*(1+2^(n+2))-3^(n+2). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008
If we define f(m,j,x)=sum(binomial(m,k)*stirling2(k,j)*x^(m-k),k=j..m) then a(n-2)=f(n,2,2), (n>=2). [From Milan R. Janjic (agnus(AT)blic.net), Apr 26 2009]
E.g.f.: diff(exp(2*x)*((exp(x)-1)^2)/2!,x$2). See the Sheffer comment given above. [From Wolfdieter Lang, Oct 08 2011]
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MAPLE
| with(combinat):a:=n->stirling2(n, 4)-stirling2(n-1, 4): seq(a(n), n=4..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
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CROSSREFS
| Equals (1/2) A038721(n+1). First differences of A000453. Partial sums of A027650. Pairwise sums of A099110. Odd part of A019333.
Cf. A000392, A032263, A000079, A001047.
Sequence in context: A145875 A068970 A141530 * A005770 A030053 A072844
Adjacent sequences: A016266 A016267 A016268 * A016270 A016271 A016272
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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