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%I #19 Sep 14 2019 16:41:00
%S 1,1,1,-1,0,1,1,1,0,1,-1,0,0,0,1,-1,-1,1,0,0,1,-1,0,0,0,0,0,1,1,1,0,1,
%T 0,0,0,1,0,0,-1,0,0,0,0,0,1,-1,-1,0,0,1,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,
%U 0,1,-1,-1,1,-1,0,1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,1
%N A129360 * A115361.
%C Row sums = A104117: (1, 2, 0, 3, 0, 0, 0, 4, 0, 0, ...).
%H Andrew Howroyd, <a href="/A103994/b103994.txt">Table of n, a(n) for n = 1..1275</a>
%F A129360 * A115361 as infinite lower triangular matrices.
%F T(n,k) = A209635(n/k) for k | n, T(n,k) = 0 otherwise. - _Andrew Howroyd_, Aug 03 2018
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e -1, 0, 1;
%e 1, 1, 0, 1;
%e -1, 0, 0, 0, 1;
%e -1, -1, 1, 0, 0, 1;
%e -1, 0, 0, 0, 0, 0, 1;
%e 1, 1, 0, 1, 0, 0, 0, 1;
%e ...
%t T[n_, k_] := If[Divisible[n, k], MoebiusMu[(n/k)/2^IntegerExponent[n/k, 2]], 0];
%t Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 14 2019 *)
%o (PARI) tabl(nn) = {Tm = matrix(nn, nn, n, k, if (! (n % k), moebius(n/k), 0)); Tr = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = Tr^(-1); Tp = Tm*Ti*Ti; for (n=1, nn, for (k=1, n, print1(Tp[n, k], ", ");); print(););}
%o (PARI) T(n, k)={ if(n%k, 0, sumdiv(n/k, d, my(e=valuation(d, 2)); if(d==1<<e, moebius(n/(k*d))*(1 + e), 0))) } \\ _Andrew Howroyd_, Aug 03 2018
%Y Column 1 is A209635 (Moebius transform of A104117).
%Y Row sums are A104117.
%Y Cf. A129360, A115361.
%K tabl,sign
%O 1,1
%A _Gary W. Adamson_, Apr 15 2007
%E More terms from _Michel Marcus_, Mar 28 2015
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