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A334312 Triangle read by rows: T(n,k) = Sum_{i=k..n} A191898(i,k).

%I A334312 #32 May 17 2020 13:51:32
%S A334312 1,2,-1,3,0,-2,4,-1,-1,-1,5,0,0,0,-4,6,-1,-2,-1,-3,2,7,0,-1,0,-2,3,-6,
%T A334312 8,-1,0,-1,-1,2,-5,-1,9,0,-2,0,0,0,-4,0,-2,10,-1,-1,-1,-4,-1,-3,-1,-1,
%U A334312 4,11,0,0,0,-3,0,-2,0,0,5,-10,12,-1,-2,-1,-2,2,-1,-1,-2,4,-9,2
%N A334312 Triangle read by rows: T(n,k) = Sum_{i=k..n} A191898(i,k).
%C A334312 A334314(n)/A334313(n) = Sum_{k=1..n} T(n,k)/k.
%H A334312 Mats Granvik, <a href="/A334312/a334312.txt">Mathematica program for recurrence 1</a>
%H A334312 Mats Granvik, <a href="/A334312/a334312_1.txt">Mathematica program for recurrence 2</a>
%F A334312 Let: f(n) = Sum_{ d divides n } d*mu(d) = A023900(n), then T(n,k) = Sum_{i=k..n} f(gcd(i,k)).
%F A334312 Recurrence 1:
%F A334312 T(n, 1) = n.
%F A334312 T(n, k) = [n >= k]*[k > 1]*(Sum_{j=0..n-k} Sum_{i=j+1..k-1} (T(k-1,i)-T(k,i)) -Sum_{i=n-k+1..n-1} T(i, k)).
%F A334312 Recurrence 2:
%F A334312 T(n, 1) = n.
%F A334312 T(n, k) = [n >= k]*(Sum_{i=n-k+1..k-1}T(k-1,i)-T(k,i)) + [n >= 2*k]*T(n-k,k).
%e A334312 Triangle begins:
%e A334312 1,
%e A334312 2,  -1,
%e A334312 3,   0,  -2,
%e A334312 4,  -1,  -1,  -1,
%e A334312 5,   0,   0,   0,  -4,
%e A334312 6,  -1,  -2,  -1,  -3,   2,
%e A334312 7,   0,  -1,   0,  -2,   3,  -6,
%e A334312 8,  -1,   0,  -1,  -1,   2,  -5,  -1,
%e A334312 9,   0,  -2,   0,   0,   0,  -4,   0,  -2,
%e A334312 10, -1,  -1,  -1,  -4,  -1,  -3,  -1,  -1,   4,
%e A334312 11,  0,   0,   0,  -3,   0,  -2,   0,   0,   5,  -10,
%e A334312 12, -1,  -2,  -1,  -2,   2,  -1,  -1,  -2,   4,   -9,   2,
%e A334312 ...
%t A334312 nn=14; f[n_] := Total[Divisors[n]*MoebiusMu[Divisors[n]]]; Flatten[Table[Table[Sum[f[GCD[i, k]], {i, k, n}], {k, 1, n}], {n, 1, nn}]]
%Y A334312 Row sums give A000012.
%Y A334312 Cf. A191898, A309229, A023900.
%K A334312 sign,tabl
%O A334312 1,2
%A A334312 _Mats Granvik_, Apr 22 2020