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