A175685 Array a(n,m) = Sum_{j=floor((n-1)/2)-m..floor(n-1)/2} binomial(n-j-1,j) read by antidiagonals.

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 3, 4, 3, 2, 1, 1, 1, 7, 5, 3, 2, 1, 1, 4, 7, 8, 5, 3, 2, 1, 1, 1, 14, 12, 8, 5, 3, 2, 1, 1

Offset: 1

Author

Roger L. Bagula, Dec 04 2010

Keywords

• nonn
• tabl
• easy

Comments

A102426 defines an array of binomials in which partial sums of row n yield row a(n,.).

Examples

			a(n,m) starts in row n=1 as
  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
  1,  4,  5,  5,  5,  5,  5,  5,  5,  5,  5, ...
  3,  7,  8,  8,  8,  8,  8,  8,  8,  8,  8, ...
  1,  7, 12, 13, 13, 13, 13, 13, 13, 13, 13, ...
  4, 14, 20, 21, 21, 21, 21, 21, 21, 21, 21, ...
  1, 11, 26, 33, 34, 34, 34, 34, 34, 34, 34, ...
		

References

• Burton, David M., Elementary number theory, McGraw Hill, N.Y., 2002, p. 286.

Crossrefs

Cf. A000045, A011973, A102426.

Programs

Maple

A175685 := proc(n,m) upl := floor( (n-1)/2) ; add( binomial(n-j-1,j),j=upl-m .. upl) ; end proc: # R. J. Mathar, Dec 05 2010

Mathematica

a = Table[Table[Sum[Binomial[n -j - 1, j], {j, Floor[(n - 1)/2] - m, Floor[(n - %t 1)/2]}], {n, 0, 10}], {m, 0, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}];Flatten[%]