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

0,1

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

Links

Table of n, a(n) for n=0..104.

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

Example

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;

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 *)

Prog

(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<<e)}
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Aug 03 2018

Crossrefs

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

Cf. A036987, A209229.

Sequence in context: A014039 A016410 A230412 * A115358 A325898 A117904

Adjacent sequences: A115358 A115359 A115360 * A115362 A115363 A115364

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 21 2006

Status

approved