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 *)
With[{nn=30}, CoefficientList[Series[(Exp[x]-x-1)^2, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, May 29 2023 *)
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<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
CROSSREFS
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