The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A188440 Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}. 2
 1, 1, 1, 2, 1, 2, 1, 4, 4, 1, 4, 4, 1, 6, 12, 8, 1, 6, 12, 8, 1, 8, 24, 32, 16, 1, 8, 24, 32, 16, 1, 10, 40, 80, 80, 32, 1, 10, 40, 80, 80, 32, 1, 12, 60, 160, 240, 192, 64, 1, 12, 60, 160, 240, 192, 64, 1, 14, 84, 280, 560, 672, 448, 128, 1, 14, 84, 280 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A subset S of {1,2,...,n} is antisymmetric if x is an element of S implies n+1-x is not an element of S. In other words, the sum of any two elements of S does not equal n+1. For example, {1,2,5} is an antisymmetric subset of {1,2,3,4,5,6,7}. If n is odd, (n+1)/2 cannot be an element of an antisymmetric subset of {1,2,...,n}. (Note that for n=0, we define {1,...,n} to be the empty set, and thus T(0,0)=1 since the empty set is vacuously antisymmetric.) We note, for example, that T(100,k) provides the number of possible size-k committees of the U.S. Senate in which no two members are from the same state. Triangle, read by rows, A013609 rows repeated. - Philippe Deléham, Apr 09 2012 Triangle, with zeros omitted, given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012 LINKS T. D. Noe, Rows n = 0..100, flattened Dennis Walsh, Notes on antisymmetric subsets of {1,2,...,n} FORMULA T(n,k) = 2^k*C(floor(n/2),k) where C(*,*) denotes a binomial coefficient. Sum(T(n,k),k=0..floor(n/2)) = 3^floor(n/2) = A108411(n). G.f. for columns(k fixed):(2t^2)^k/((1-t)*(1-t^2)^k). T(n,k) = A152198(n,k)*2^k. - Philippe Deléham, Apr 09 2012 G.f.: (1+x)/(1-x^2-2*y*x^2). - Philippe Deléham, Apr 09 2012 T(n,k) = T(n-2,k) + 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n.- Philippe Deléham, Apr 09 2012 EXAMPLE Triangle T(n,k) initial values 0 <= k <= floor(n/2), n=0..13: 1 1 1 2 1 2 1 4 4 1 4 4 1 6 12 8 1 6 12 8 1 8 24 32 16 1 8 24 32 16 1 10 40 80 80 32 1 10 40 80 80 32 1 12 60 160 240 192 64 1 12 60 160 240 192 64 ... For n=7 and k=2, T(7,2)=12 since there are 12 antisymmetric size-2 subsets of {1,2,...,7}: {1,2}, {1,3}, {1,5}, {1,6}, {2,3}, {2,5}, {2,7}, {3,6}, {3,7}, {5,6}, {5,7}, and {6,7}. (1, 0, -1, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, ...) begins: 1 1 0 1 2 0 1 2 0 0 1 4 4 0 0 1 4 4 0 0 0 1 6 12 8 0 0 0 1 6 12 8 0 0 0 0 1 8 24 32 16 0 0 0 0 1 8 24 32 16 0 0 0 0 0 1 10 40 80 80 32 0 0 0 0 0 1 10 40 80 80 32 0 0 0 0 0 0 1 12 60 160 240 192 64 0 0 0 0 0 0 1 12 60 160 240 192 64 0 0 0 0 0 0 0 - Philippe Deléham, Apr 09 2012 MAPLE seq(seq(binomial(floor(n/2), k)*2^k, k=0..floor(n/2)), n=0..22); MATHEMATICA Table[ CoefficientList[(1 + 2*x)^n, x] , {n, 0, 7}, {2}] // Flatten (* Jean-François Alcover, Aug 19 2013, after Philippe Deléham *) CROSSREFS Cf. A108411, row sums of triangle T(n,k). Cf. A000079, A007318, A013109, A152198 Sequence in context: A157333 A002852 A266081 * A216327 A099875 A079499 Adjacent sequences: A188437 A188438 A188439 * A188441 A188442 A188443 KEYWORD nice,easy,nonn,tabf AUTHOR Dennis P. Walsh, Mar 31 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 29 22:10 EDT 2023. Contains 365781 sequences. (Running on oeis4.)