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

0,14

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

Links

Table of n, a(n) for n=0..100.

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

Example

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; ...

Crossrefs

Sequence in context: A281244 A284585 A280456 * A350847 A026821 A377334

Adjacent sequences: A103630 A103631 A103632 * A103634 A103635 A103636

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 11 2005

Status

approved