

A052515


Number of ordered pairs of complementary subsets of an nset with both subsets of cardinality at least 2.


9



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 nth 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 doublysurjective 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 nset 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 doublysurjective 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(n1).  Ralf Stephan, Jan 11 2004
G.f.: 2*x^4*(32*x)/((1x)^2*(12*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((64*x)/(1x)^2/(12*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)x1)^2))) \\ Joerg Arndt, Apr 10 2013
(MAGMA) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (Exp(x)1x)^2 )); [0, 0, 0, 0] cat [Factorial(n+3)*b[n]: n in [1..m5]]; // G. C. Greubel, May 13 2019
(Sage) (2*x^4*(32*x)/((1x)^2*(12*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



