A103633 Triangle read by rows: triangle of repeated stepped binomial coefficients.

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 0, 0, 0, 0, 0, 1, 6, 15

Offset: 0

Author

Paul Barry, Feb 11 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums are Sum_{k=0..n} binomial(floor(n/2),n-k) = (1,1,2,2,4,4,...). Diagonal sums have g.f. (1+x^2)/(1-x^3-x^4) (see A079398). Matrix inverse of the signed triangle (-1)^(n-k)T(n,k) is A103631. Matrix inverse of T(n,k) is the alternating signed version of A103631.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ....] DELTA [1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 08 2005

Examples

			Triangle begins:
  1;
  0,  1;
  0,  1,  1;
  0,  0,  1,  1;
  0,  0,  1,  2,  1;
  0,  0,  0,  1,  2,  1;
  0,  0,  0,  1,  3,  3,  1;
  0,  0,  0,  0,  1,  3,  3,  1;
  0,  0,  0,  0,  1,  4,  6,  4,  1;
  0,  0,  0,  0,  0,  1,  4,  6,  4,  1;
  0,  0,  0,  0,  0,  1,  5, 10, 10,  5,  1;
  0,  0,  0,  0,  0,  0,  1,  5, 10, 10,  5,  1;
  0,  0,  0,  0,  0,  0,  1,  6, 15, 20, 15,  6,  1; ...
		

Formula

Number triangle T(n, k) = binomial(floor(n/2), n-k).

Sum_{n>=0} T(n, k) = A000045(k+2) = Fibonacci(k+2). - Philippe Deléham, Oct 08 2005

Sum_{k=0..n} T(n,k) = 2^floor(n/2) = A016116(n). - Philippe Deléham, Dec 03 2006

G.f.: (1+x*y)/(1-x^2*y-x^2*y^2). - Philippe Deléham, Nov 10 2013

T(n,k) = T(n-2,k-1) + T(n-2,k-2) for n > 2, T(0,0) = T(,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Nov 10 2013