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A344479 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} gcd(x_1, x_2, ..., x_k).

%I A344479 #40 Jun 11 2021 11:15:40
%S A344479 1,1,3,1,5,6,1,9,12,10,1,17,30,24,15,1,33,84,76,37,21,1,65,246,276,
%T A344479 141,61,28,1,129,732,1060,649,267,80,36,1,257,2190,4164,3165,1417,400,
%U A344479 112,45,1,513,6564,16516,15697,8091,2528,624,145,55,1,1025,19686,65796,78261,47521,17128,4432,885,189,66
%N A344479 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} gcd(x_1, x_2, ..., x_k).
%H A344479 Seiichi Manyama, <a href="/A344479/b344479.txt">Antidiagonals n = 1..140, flattened</a>
%F A344479 G.f. of column k: (1/(1 - x)) * Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^k.
%F A344479 T(n,k) = Sum_{j=1..n} phi(j) * floor(n/j)^k.
%e A344479 G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
%e A344479 Square array begins:
%e A344479    1,  1,   1,    1,    1,     1, ...
%e A344479    3,  5,   9,   17,   33,    65, ...
%e A344479    6, 12,  30,   84,  246,   732, ...
%e A344479   10, 24,  76,  276, 1060,  4164, ...
%e A344479   15, 37, 141,  649, 3165, 15697, ...
%e A344479   21, 61, 267, 1417, 8091, 47521, ...
%t A344479 T[n_, k_] := Sum[EulerPhi[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 A344479 (PARI) T(n, k) = sum(j=1, n, eulerphi(j)*(n\j)^k);
%Y A344479 Columns k=1..5 give A000217, A018806, A344522, A344523, A344524.
%Y A344479 T(n,n) gives A344525.
%Y A344479 Cf. A343510, A343516, A344527.
%K A344479 nonn,tabl
%O A344479 1,3
%A A344479 _Seiichi Manyama_, May 22 2021