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A344527 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of ordered k-tuples (x_1, x_2, ..., x_k) with gcd(x_1, x_2, ..., x_k) = 1 (1 <= {x_1, x_2, ..., x_k} <= n).

%I A344527 #43 Nov 07 2023 08:24:25
%S A344527 1,1,1,1,3,1,1,7,7,1,1,15,25,11,1,1,31,79,55,19,1,1,63,241,239,115,23,
%T A344527 1,1,127,727,991,607,181,35,1,1,255,2185,4031,3091,1199,307,43,1,1,
%U A344527 511,6559,16255,15559,7501,2303,439,55,1,1,1023,19681,65279,77995,45863,16531,3823,637,63,1
%N A344527 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of ordered k-tuples (x_1, x_2, ..., x_k) with gcd(x_1, x_2, ..., x_k) = 1 (1 <= {x_1, x_2, ..., x_k} <= n).
%H A344527 Seiichi Manyama, <a href="/A344527/b344527.txt">Antidiagonals n = 1..140, flattened</a>.
%F A344527 G.f. of column k: (1/(1 - x)) * Sum_{i>=1} mu(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^k.
%F A344527 T(n,k) = Sum_{j=1..n} mu(j) * floor(n/j)^k.
%F A344527 T(n,k) = n^k - Sum_{j=2..n} T(floor(n/j),k).
%e A344527 G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} mu(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
%e A344527 Square array begins:
%e A344527   1,  1,   1,    1,    1,     1, ...
%e A344527   1,  3,   7,   15,   31,    63, ...
%e A344527   1,  7,  25,   79,  241,   727, ...
%e A344527   1, 11,  55,  239,  991,  4031, ...
%e A344527   1, 19, 115,  607, 3091, 15559, ...
%e A344527   1, 23, 181, 1199, 7501, 45863, ...
%t A344527 T[n_, k_] := Sum[MoebiusMu[j] * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 22 2021 *)
%o A344527 (PARI) T(n, k) = sum(j=1, n, moebius(j)*(n\j)^k);
%o A344527 (PARI) T(n, k) = n^k-sum(j=2, n, T(n\j, k));
%o A344527 (Python)
%o A344527 from functools import lru_cache
%o A344527 from itertools import count, islice
%o A344527 @lru_cache(maxsize=None)
%o A344527 def A344527_T(n,k):
%o A344527     if n == 0:
%o A344527         return 0
%o A344527     c, j, k1 = 1, 2, n//2
%o A344527     while k1 > 1:
%o A344527         j2 = n//k1 + 1
%o A344527         c += (j2-j)*A344527_T(k1,k)
%o A344527         j, k1 = j2, n//j2
%o A344527     return n*(n**(k-1)-1)-c+j
%o A344527 def A344527_gen(): # generator of terms
%o A344527     return (A344527_T(k+1, n-k) for n in count(1) for k in range(n))
%o A344527 A344527_list = list(islice(A344527_gen(),30)) # _Chai Wah Wu_, Nov 02 2023
%Y A344527 Columns k=1..6 give A000012, A018805, A071778, A082540, A082544, A343978.
%Y A344527 T(n,n) gives A332468.
%Y A344527 Cf. A343510, A343516, A344479.
%K A344527 nonn,tabl
%O A344527 1,5
%A A344527 _Seiichi Manyama_, May 22 2021