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) * [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
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
KEYWORD
AUTHOR
Paul Barry, Jan 21 2006
STATUS
approved