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A052515
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Number of ordered pairs of complementary subsets of an n-set with both subsets of cardinality at least 2.
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9
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0, 0, 0, 0, 6, 20, 50, 112, 238, 492, 1002, 2024, 4070, 8164, 16354, 32736, 65502, 131036, 262106, 524248, 1048534, 2097108, 4194258, 8388560, 16777166, 33554380, 67108810, 134217672, 268435398, 536870852, 1073741762
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OFFSET
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0,5
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COMMENTS
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a(n) is the number of binary sequences of length n having at least two 0's and at least two 1's. a(4)=6 because there are six binary sequences of length four that have two or more 0's and two or more 1's: 0011, 0101, 0110, 1100, 1010, 1001. - Geoffrey Critzer, Feb 11 2009
For n>3, a(n) is also the sum of those terms from the n-th row of Pascal's triangle which also occur in A006987: 6, 10+10, 15+20+15, 21+35+35+21,... - Douglas Latimer, Apr 02 2012
From Dennis P. Walsh, Apr 09 2013: (Start)
Column 2 of triangle A200091.
Number of doubly-surjective functions f:[n]->[2].
Number of ways to distribute n different toys to 2 children so that each child gets at least 2 toys. (End)
a(n) is the number of subsets of an n-set of cardinality k with 2 <= k <= n - 2. - Rick L. Shepherd, Dec 05 2014
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 81
Dennis Walsh, Notes on doubly-surjective finite functions
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
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FORMULA
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E.g.f.: (exp(x) - x - 1)^2. - Joerg Arndt, Apr 10 2013
n*a(n+2) - (1+3*n)*a(n+1) + 2(1+n)*a(n) = 0, with a(0) = .. = a(3) = 0, a(4) = 6.
For n>2, a(n) = 2^n - 2n - 2 = A005803(n) - 2 = A070313(n) - 1 = A071099(n) - A071099(n+1) + 1 = 2*A000247(n-1). - Ralf Stephan, Jan 11 2004
G.f.: 2*x^4*(3-2*x)/((1-x)^2*(1-2*x)). - Colin Barker, Feb 19 2012
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MAPLE
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Pairs spec := [S, {S=Prod(B, B), B=Set(Z, 2 <= card)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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Join[{0, 0, 0}, LinearRecurrence[{4, -5, 2}, {0, 6, 20}, 35]] (* G. C. Greubel, May 13 2019 *)
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PROG
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(PARI) concat([0, 0, 0, 0], Vec((6-4*x)/(1-x)^2/(1-2*x)+O(x^35))) \\ Charles R Greathouse IV, Apr 03 2012
(PARI) x='x+O('x^35); concat([0, 0, 0, 0], Vec(serlaplace((exp(x)-x-1)^2))) \\ Joerg Arndt, Apr 10 2013
(MAGMA) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (Exp(x)-1-x)^2 )); [0, 0, 0, 0] cat [Factorial(n+3)*b[n]: n in [1..m-5]]; // G. C. Greubel, May 13 2019
(Sage) (2*x^4*(3-2*x)/((1-x)^2*(1-2*x))).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, May 13 2019
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CROSSREFS
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Sequence in context: A216175 A161409 A002415 * A067117 A267168 A266760
Adjacent sequences: A052512 A052513 A052514 * A052516 A052517 A052518
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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More terms from Ralf Stephan, Jan 11 2004
Definition corrected by Rainer Rosenthal, Feb 12 2010
Definition further clarified by Rick L. Shepherd, Dec 05 2014
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STATUS
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approved
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