A115361 Inverse of matrix (1,x)-(x,x^2) (expressed in Riordan array notation).

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

Offset: 0

Author

Paul Barry, Jan 21 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are the 'ruler function' A001511. Columns are stretched Fredholm-Rueppel sequences (A036987). Inverse is A115359.

Eigensequence of triangle A115361 = A018819 starting with offset 1: (1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, ...). - Gary W. Adamson, Nov 21 2009

From Gary W. Adamson, Nov 27 2009: (Start)

A115361 * [1, 2, 3, ...] = A129527 = (1, 3, 3, 7, 5, 9, 7, 15, ...).

(A115361)^(-1) * [1, 2, 3, ...] = A115359 * [1, 2, 3, ...] = A026741 starting /Q (1, 1, 3, 2, 5, 3, 7, 4, 9, ...). (End)

This is the lower-left triangular part of the inverse of the infinite matrix A_{ij} = [i=j] - [i=2j], its upper-right part (above / right to the diagonal) being zero. The n-th row has 1 in column n/2^i, i = 0, 1, ... as long as this is an integer. - M. F. Hasler, May 13 2018

The rows are the reversed binary expansions of A127804. - Tilman Piesk, Jun 10 2025

Examples

			Triangle begins:
  1;
  1,1;
  0,0,1;
  1,1,0,1;
  0,0,0,0,1;
  0,0,1,0,0,1;
  0,0,0,0,0,0,1;
  1,1,0,1,0,0,0,1;
  0,0,0,0,0,0,0,0,1;
  0,0,0,0,1,0,0,0,0,1;
  0,0,0,0,0,0,0,0,0,0,1;
		

Links

• Tilman Piesk, Illustration of first 32 rows.

Crossrefs

Cf. A016741, A018819, A129527. - Gary W. Adamson, Nov 21 2009

Cf. A036987, A209229, A127804.

Programs

Maple

A115361 := proc(n,k)
    for j from 0 do
        if k+(2*j-1)*(k+1) > n then
            return 0 ;
        elif k+(2^j-1)*(k+1) = n then
            return 1 ;
        end if;
    end do;
end proc: # R. J. Mathar, Jul 14 2012

Mathematica

(*recurrence*)
Clear[t]
t[1, 1] = 1;
t[n_, k_] :=
t[n, k] =
  If[k == 1, Sum[t[n, k + i], {i, 1, 2 - 1}],
   If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 14}]] (* Mats Granvik, Jun 26 2014 *)

PARI

tabl(nn) = {T = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = T^(-1); for (n=1, nn, for (k=1, n, print1(Ti[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

PARI

A115361_row(n,v=vector(n))={until(bittest(n,0)||!n\=2,v[n]=1);v} \\ Yields the n-th row (of length n). - M. F. Hasler, May 13 2018

PARI

T(n,k)={if(n%k, 0, my(e=valuation(n/k,2)); n/k==1<Andrew Howroyd, Aug 03 2018

Python

# translation of Maple code by R. J. Mathar
def a115361(n, k):
    j = 0
    while True:
        if k + (2*j - 1) * (k + 1) > n:
            return 0
        elif k + (2**j - 1) * (k + 1) == n:
            return 1
        else:
            j += 1  #  Tilman Piesk, Jun 10 2025

Formula

Number triangle whose k-th column has g.f. x^k*sum{j>=0} x^((2^j-1)*(k+1)).

T(n,k) = A209229((n+1)/(k+1)) for k+1 divides n+1, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 05 2018