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

0,1

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

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.

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

Example

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)

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 *)

Prog

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

Crossrefs

Sequence in context: A116938 A105589 A359578 * A167374 A294821 A132971

Adjacent sequences: A097803 A097804 A097805 * A097807 A097808 A097809

Keyword

easy,nonn,tabl

Author

Paul Barry, Aug 25 2004

Status

approved