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%I #42 Sep 08 2022 08:45:14
%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 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, 15}] // Flatten (* or *)
%t Table[Floor[(# +2)/(n+2)] & /@ Range[n+1], {n, 0, 15}] // Flatten (* _Michael De Vlieger_, Jul 21 2016 *)
%o (PARI) T(n, k) = if(k==n || k==n-1, 1, 0); \\ _G. C. Greubel_, Jul 11 2019
%o (Magma) [k eq n or k eq n-1 select 1 else 0: k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jul 11 2019
%o (Sage)
%o def T(n, k):
%o if (k==n or k==n-1): return 1
%o else: return 0
%o [[T(n, k) for k in (0..n)] for n in (0..15)] # _G. C. Greubel_, Jul 11 2019
%K easy,nonn,tabl
%O 0,1
%A _Paul Barry_, Aug 25 2004
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