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A103633 Triangle read by rows: triangle of repeated stepped binomial coefficients. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A026821 A039964 A340655

Adjacent sequences:  A103630 A103631 A103632 * A103634 A103635 A103636

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Feb 11 2005

STATUS

approved

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Last modified April 20 06:52 EDT 2021. Contains 343121 sequences. (Running on oeis4.)