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

%I A115361 #38 Jul 16 2025 17:10:06
%S A115361 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,
%T A115361 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,
%U A115361 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
%N A115361 Inverse of matrix (1,x)-(x,x^2) (expressed in Riordan array notation).
%C A115361 Row sums are the 'ruler function' A001511. Columns are stretched Fredholm-Rueppel sequences (A036987). Inverse is A115359.
%C A115361 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
%C A115361 From _Gary W. Adamson_, Nov 27 2009: (Start)
%C A115361 A115361 * [1, 2, 3, ...] = A129527 = (1, 3, 3, 7, 5, 9, 7, 15, ...).
%C A115361 (A115361)^(-1) * [1, 2, 3, ...] = A115359 * [1, 2, 3, ...] = A026741 starting /Q (1, 1, 3, 2, 5, 3, 7, 4, 9, ...). (End)
%C A115361 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
%C A115361 The rows are the reversed binary expansions of A127804. - _Tilman Piesk_, Jun 10 2025
%H A115361 Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:Sequence_A127804_from_binary_triangle.svg">Illustration of first 32 rows</a>.
%F A115361 Number triangle whose k-th column has g.f. x^k*sum{j>=0} x^((2^j-1)*(k+1)).
%F A115361 T(n,k) = A209229((n+1)/(k+1)) for k+1 divides n+1, T(n,k) = 0 otherwise. - _Andrew Howroyd_, Aug 05 2018
%e A115361 Triangle begins:
%e A115361   1;
%e A115361   1,1;
%e A115361   0,0,1;
%e A115361   1,1,0,1;
%e A115361   0,0,0,0,1;
%e A115361   0,0,1,0,0,1;
%e A115361   0,0,0,0,0,0,1;
%e A115361   1,1,0,1,0,0,0,1;
%e A115361   0,0,0,0,0,0,0,0,1;
%e A115361   0,0,0,0,1,0,0,0,0,1;
%e A115361   0,0,0,0,0,0,0,0,0,0,1;
%p A115361 A115361 := proc(n,k)
%p A115361     for j from 0 do
%p A115361         if k+(2*j-1)*(k+1) > n then
%p A115361             return 0 ;
%p A115361         elif k+(2^j-1)*(k+1) = n then
%p A115361             return 1 ;
%p A115361         end if;
%p A115361     end do;
%p A115361 end proc: # _R. J. Mathar_, Jul 14 2012
%t A115361 (*recurrence*)
%t A115361 Clear[t]
%t A115361 t[1, 1] = 1;
%t A115361 t[n_, k_] :=
%t A115361 t[n, k] =
%t A115361   If[k == 1, Sum[t[n, k + i], {i, 1, 2 - 1}],
%t A115361    If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
%t A115361 Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 14}]] (* _Mats Granvik_, Jun 26 2014 *)
%o A115361 (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
%o A115361 (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
%o A115361 (PARI) T(n,k)={if(n%k, 0, my(e=valuation(n/k,2)); n/k==1<<e)}
%o A115361 for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ _Andrew Howroyd_, Aug 03 2018
%o A115361 (Python)
%o A115361 # translation of Maple code by R. J. Mathar
%o A115361 def a115361(n, k):
%o A115361     j = 0
%o A115361     while True:
%o A115361         if k + (2*j - 1) * (k + 1) > n:
%o A115361             return 0
%o A115361         elif k + (2**j - 1) * (k + 1) == n:
%o A115361             return 1
%o A115361         else:
%o A115361             j += 1  #  _Tilman Piesk_, Jun 10 2025
%Y A115361 Cf. A016741, A018819, A129527. - _Gary W. Adamson_, Nov 21 2009
%Y A115361 Cf. A036987, A209229, A127804.
%K A115361 easy,nonn,tabl
%O A115361 0,1
%A A115361 _Paul Barry_, Jan 21 2006