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 A097806 Riordan array (1+x,1) read by rows. 45

%I

%S 1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,

%T 1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0,

%U 0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1

%N Riordan array (1+x,1) read by rows.

%C Pair sum operator. Columns have g.f. (1+x)x^k. Row sums are A040000. Diagonal sums are (1,1,1,....). Riordan inverse is (1/(1+x), 1). A097806=B*A059260^(-1), where B is the binomial matrix.

%C Triangle T(n,k), 0<=k<=n, read by rows given by [1, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, May 01 2007

%C Table T(n,k) read by antidiagonals. T(n,1) = 1, T(n,2) = 1, T(n,k) = 0, k > 2. - _Boris Putievskiy_, Jan 17 2013

%H Michael De Vlieger, <a href="/A097806/b097806.txt">Table of n, a(n) for n = 0..10010</a> (Rows 0 <= n <= 140)

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.

%F T(n, k) = if(n=k or n-k=1, 1, 0).

%F a(n) = A103451(n+1). - _Philippe Deléham_, Oct 16 2007

%F From _Boris Putievskiy_, Jan 17 2013: (Start)

%F a(n) = floor((A002260(n)+2)/(A003056(n)+2)), n > 0.

%F a(n) = floor((i+2)/(t+2)), n > 0,

%F where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)

%F G.f.: (1+x)/(1-x*y). - _R. J. Mathar_, Aug 11 2015

%e Rows begin {1}, {1,1}, {0,1,1}, {0,0,1,1}...

%e From _Boris Putievskiy_, Jan 17 2013: (Start)

%e The start of the sequence as table:

%e 1..1..0..0..0..0..0...

%e 1..1..0..0..0..0..0...

%e 1..1..0..0..0..0..0...

%e 1..1..0..0..0..0..0...

%e 1..1..0..0..0..0..0...

%e 1..1..0..0..0..0..0...

%e 1..1..0..0..0..0..0...

%e . . .

%e The start of the sequence as triangle array read by rows:

%e 1;

%e 1,1;

%e 0,1,1;

%e 0,0,1,1;

%e 0,0,0,1,1;

%e 0,0,0,0,1,1;

%e 0,0,0,0,0,1,1;

%e 0,0,0,0,0,0,1,1;

%e . . .

%e Row number r (r>4) contains (r-2) times '0' and 2 times '1'. (End)

%p A097806 := proc(n,k)

%p if k =n or k=n-1 then

%p 1;

%p else

%p 0;

%p end if;

%p end proc: # _R. J. Mathar_, Jun 20 2015

%t Table[Boole[n <= # <= n + 1] & /@ Range[n + 1], {n, 0, 14}] // Flatten (* or *)

%t Table[Floor[(# + 2)/(n + 2)] & /@ Range[n + 1], {n, 0, 14}] // Flatten (* _Michael De Vlieger_, Jul 21 2016 *)

%K easy,nonn,tabl

%O 0,1

%A _Paul Barry_, Aug 25 2004

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