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 A052515 Number of ordered pairs of complementary subsets of an n-set with both subsets of cardinality at least 2. 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 LINKS 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). FORMULA 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 MAPLE Pairs spec := [S, {S=Prod(B, B), B=Set(Z, 2 <= card)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA Join[{0, 0, 0}, LinearRecurrence[{4, -5, 2}, {0, 6, 20}, 35]] (* G. C. Greubel, May 13 2019 *) PROG (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:=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 CROSSREFS Sequence in context: A216175 A161409 A002415 * A067117 A267168 A266760 Adjacent sequences:  A052512 A052513 A052514 * A052516 A052517 A052518 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 EXTENSIONS 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 STATUS approved

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Last modified August 9 12:53 EDT 2022. Contains 356026 sequences. (Running on oeis4.)