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

1,7

Comments

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

References

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

Links

Table of n, a(n) for n=1..45.

Example

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, ...

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[%]

Crossrefs

Cf. A000045, A011973, A102426.

Sequence in context: A349670 A086275 A066855 * A331182 A182591 A321393

Adjacent sequences: A175682 A175683 A175684 * A175686 A175687 A175688

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Dec 04 2010

Status

approved