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

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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Author

Paul Barry, Aug 25 2004

Keywords

• easy
• nonn
• tabl

Comments

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.

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

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

Examples

			Rows begin {1}, {1,1}, {0,1,1}, {0,0,1,1}...
From _Boris Putievskiy_, Jan 17 2013: (Start)
The start of the sequence as table:
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
. . .
The start of the sequence as triangle array read by rows:
  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, 1, 1; . . .
Row number r (r>4) contains (r-2) times '0' and 2 times '1'. (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10010 (Rows 0 <= n <= 140)
• Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Programs

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

Maple

A097806 := proc(n,k)
    if k =n or k=n-1 then
        1;
    else
        0;
    end if;
end proc: # R. J. Mathar, Jun 20 2015

Mathematica

Table[Boole[n <= # <= n+1] & /@ Range[n+1], {n, 0, 15}] // Flatten (* or *)
Table[Floor[(# +2)/(n+2)] & /@ Range[n+1], {n, 0, 15}] // Flatten (* Michael De Vlieger, Jul 21 2016 *)

PARI

T(n, k) = if(k==n || k==n-1, 1, 0); \\ G. C. Greubel, Jul 11 2019

Sage

def T(n, k):
    if (k==n or k==n-1): return 1
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Jul 11 2019

Formula

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

a(n) = A103451(n+1). - Philippe Deléham, Oct 16 2007

From Boris Putievskiy, Jan 17 2013: (Start)

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

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

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

G.f.: (1+x)/(1-x*y). - R. J. Mathar, Aug 11 2015