cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A109267 Riordan array (1/(1 - x*c(x) - x^2*c(x)^2), x*c(x)) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 9, 6, 3, 1, 29, 19, 10, 4, 1, 97, 63, 34, 15, 5, 1, 333, 215, 118, 55, 21, 6, 1, 1165, 749, 416, 201, 83, 28, 7, 1, 4135, 2650, 1485, 736, 320, 119, 36, 8, 1, 14845, 9490, 5355, 2705, 1220, 484, 164, 45, 9, 1, 53791, 34318, 19473, 9983, 4628, 1923, 703, 219, 55, 10, 1
Offset: 0

Views

Author

Paul Barry, Jun 24 2005

Keywords

Comments

Inverse of Riordan array (1-x-x^2, x(1-x)), A109264. Row sums are A109262(n+1). Diagonal sums are A109268. Columns include A081696, A109262, A109263.

Examples

			Rows begin
   1;
   1,  1;
   3,  2,  1;
   9,  6,  3,  1;
  29, 19, 10,  4,  1;
  97, 63, 34, 15,  5,  1;

Links

Crossrefs

Row sums A109262, sums along shallow diagonals A109268, A081696 (column 0), A109262 (column 1), A109263 (column 2).
Cf. A000108, A109264.

Programs

  • GAP
    Flat(List([0..10],n->List([0..n],k->Sum([0..n-k],i->(Fibonacci(i+1)-2*Fibonacci(i))*Binomial(2*n-k-i,n))))); # Muniru A Asiru, Feb 19 2018
  • Maple
    A109267 := (n,k) -> add(-combinat:-fibonacci(i-2)*binomial(2*n-k-i,n), i=0..n-k):
seq(seq(A109267(n, k), k = 0..n), n = 0..10); # Peter Bala, Feb 18 2018
  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);
RiordanArray[1/(1 - # c[#] - #^2 c[#]^2)&, # c[#]&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Formula

The production matrix M (deleting the zeros) is:
1, 1;
2, 1, 1;
2, 1, 1, 1;
2, 1, 1, 1, 1;
... such that the n-th row of the triangle is the top row of M^n. - Gary W. Adamson, Feb 16 2012
From Peter Bala, Feb 18 2018: (Start)
T(n,k) = Sum_{i = 0..n-k} (Fibonacci(i+1) - 2*Fibonacci(i))* binomial(2*n-k-i,n), 0 <= k <= n.
The n-th row polynomial of the row reverse triangle equals the n-th degree Taylor polynomial of the function (1 - 2*x)/((1 - x)*(1 - x - x^2)) * 1/(1 - x)^n about 0. For example, for n = 4, (1 - 2*x)/((1 - x)*(1 - x - x^2)) * 1/(1 - x)^4 = 1 + 4*x + 10*x^2 + 19*x^3 + 29*x^4 + O(x^5), giving (29, 19, 10, 4, 1) as row 4. (End)

A204842 Triangle by rows relating to A081696.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 9, 12, 6, 2, 29, 38, 20, 8, 2, 97, 126, 68, 30, 10, 2, 333, 430, 236, 110, 42, 12, 2, 1165, 1498, 832, 402, 166, 56, 14, 2, 4135, 5300, 2970, 1472, 640, 238, 72, 16, 2, 14845, 18980, 10710, 5410, 2440, 968, 328, 90, 18, 2
Offset: 0

Views

Author

Gary W. Adamson, Jan 19 2012

Keywords

Examples

			First few rows of the triangle =
1;
1, 2;
3, 4, 2;
9, 12, 6, 2;
29, 38, 20, 8, 2;
97, 126, 68, 30, 10, 2;
333, 430, 236, 110, 42, 12, 2;
1165, 1498, 832, 402, 166, 56, 14, 2;
4135, 5300, 2970, 1472, 640, 238, 72, 16, 2;
14845, 18980, 10710, 5410, 2440, 968, 328, 90, 18, 2;
...
Top row of M^3 = [9, 12, 6, 2, 0, 0, 0,...]

Crossrefs

Cf. A081696 (first column and also row sums).

Programs

  • Maple
    A204842T := proc(n,k)
    if n =0 and k =1 then
        2;
    elif k <0 or k >n+1 then
        0;
    else
        1;
    end if ;
end proc:
A204842 := proc(n,k)
    local A;
    A := Matrix(n+1,n+1) ;
    for row from 1 to n+1 do
    for col from 1 to n+1 do
            A[row, col] := A204842T(row-1,col-1) ;
    end do:
    end do:
    Mn := LinearAlgebra[MatrixPower](A , n);
    Mn[1,k+1] ;
end proc:
for n from 0 to 10 do
    for k from 0 to n do
    printf("%d ",A204842(n,k)) ;
    end do:
    printf("\n") ;
end do:

Formula

n-th row of the triangle is the top row of M^n, where M is the following infinite square production matrix:
1, 2, 0, 0, 0, 0,...
1, 1, 1, 0, 0, 0,...
1, 1, 1, 1, 0, 0,...
1, 1, 1, 1, 1, 0,...
1, 1, 1, 1, 1, 1,...
...
Conjecture: T(n,1) = 2*A109262(n). T(n,2)=2*A109263(n). - R. J. Mathar, Jul 21 2015
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