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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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68
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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, 35090233104309, 140455067207455, 562102681589085, 2249257981411351
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OFFSET
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0,2
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COMMENTS
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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, 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, May 13 2003
Let P(A) be the power set of an n-element set A. Then a(n-2) is 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, Jan 10 2008
a(n) also gives 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. - Wolfdieter Lang, Oct 08 2011
a(n) is also the number of even binomial coefficients in rows 0 through 2^(n+1)-1 of Pascal's triangle. - Aaron Meyerowitz, Oct 29 2013
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,2).
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LINKS
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FORMULA
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G.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)).
a(n) = (2^n)*(2^n - 1)/2 - 3^n + 2^n.
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3). - Ross La Haye, Jan 10 2008
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,2), (n >= 2). - Milan Janjic, Apr 26 2009
E.g.f.: (d^2/dx^2) (exp(2*x)*((exp(x)-1)^2)/2!). See the Sheffer comment given above. - Wolfdieter Lang, Oct 08 2011
a(n) = binomial(2^n,2) - (3^n - 2^n). - Ross La Haye, Jan 26 2016
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MAPLE
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with(combinat):a:=n->stirling2(n, 4)-stirling2(n-1, 4): seq(a(n), n=4..24); # Zerinvary Lajos, Oct 05 2007
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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