A226197 Numbers of vectors with 2*n integers such that each element is either 1 or -1, and their sum > n.

1, 1, 7, 9, 56, 79, 470, 697, 4048, 6196, 35443, 55455, 313912, 499178, 2804012, 4514873, 25211936, 40999516, 227881004, 373585604, 2068564064, 3414035527, 18844224462, 31278197839, 172186125456, 287191809724, 1577401391626, 2642070371194, 14483100716176, 24347999094724

Offset: 1

Author

Alex Ratushnyak, May 31 2013

Keywords

• nonn

Examples

			With n=3 there are 7 vectors with sum bigger than 3:
{1, 1, 1, 1, 1, 1}
{-1, 1, 1, 1, 1, 1}
{1, -1, 1, 1, 1, 1}
{1, 1, -1, 1, 1, 1}
{1, 1, 1, -1, 1, 1}
{1, 1, 1, 1, -1, 1}
{1, 1, 1, 1, 1, -1}
So a(3) = 7.
		

Programs

C

#include 
long long count, n;
void addOne(long long sum, long long added) {
  if (added==n*2) {
    if (sum>n) ++count;
    return;
  }
  ++added;
  addOne(sum+1, added);
  addOne(sum-1, added);
}
int main() {
  for (n=1; n<99; n++) {
    count = 0;
    addOne(0, 0);
    printf("%llu, ", count);
  }
  return 0;
}

Maple

A226197 := proc(n)
    add( A094527(n,k),k=1+floor(n/2)..n) ;
end proc: # R. J. Mathar, Jun 04 2013

Mathematica

a[n_] := Sum[(2*n)!/((n-k)!*(n+k)!), {k, 1 + Floor[n/2], n}]; Array[a,30] (* Giovanni Resta, May 31 2013 *)

Formula

a(n) = sum_{k = 1+floor(n/2)...n} binomial(2n,n-k), sum of the n/2 rightmost elements of row n of A094527. - Giovanni Resta, May 31 2013