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%I #46 Jan 29 2024 11:02:29
%S 1,1,3,1,5,5,1,9,11,8,1,17,29,22,9,1,33,83,74,29,15,1,65,245,274,129,
%T 55,13,1,129,731,1058,629,261,55,20,1,257,2189,4162,3129,1411,349,92,
%U 21,1,513,6563,16514,15629,8085,2407,596,105,27,1,1025,19685,65794,78129,47515,16813,4388,789,145,21
%N Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{j=1..n} gcd(j, n)^k.
%H Seiichi Manyama, <a href="/A343510/b343510.txt">Antidiagonals n = 1..140, flattened</a>
%H Peter Bala, <a href="/A368743/a368743.pdf">GCD sum theorems. Two Multivariable Cesaro Type Identities</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a> and <a href="http://mathworld.wolfram.com/EulersNumberTriangle.html">Euler's Number Triangle</a>
%F G.f. of column k: Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^(k+1).
%F T(n,k) = Sum_{d|n} phi(n/d) * d^k.
%F T(n,k) = Sum_{d|n} mu(n/d) * d * sigma_{k-1}(d).
%F Dirichlet g.f. of column k: zeta(s-1) * zeta(s-k) / zeta(s). - _Ilya Gutkovskiy_, Apr 18 2021
%F T(n,k) = Sum_{j=1..n} (n/gcd(n,j))^k*phi(gcd(n,j))/phi(n/gcd(n,j)). - _Richard L. Ollerton_, May 10 2021
%F T(n,k) = Sum_{1 <= j_1, j_2, ..., j_k <= n} gcd(j_1, j_2, ..., j_k)^2 = Sum_{d divides n} d * J_k(n/d), where J_k(n) denotes the k-th Jordan totient function. - _Peter Bala_, Jan 29 2024
%e G.f. of column 3: Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^4.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 3, 5, 9, 17, 33, 65, 129, ...
%e 5, 11, 29, 83, 245, 731, 2189, ...
%e 8, 22, 74, 274, 1058, 4162, 16514, ...
%e 9, 29, 129, 629, 3129, 15629, 78129, ...
%e 15, 55, 261, 1411, 8085, 47515, 282381, ...
%e 13, 55, 349, 2407, 16813, 117655, 823549, ...
%t T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * #^k &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Apr 18 2021 *)
%o (PARI) T(n, k) = sum(j=1, n, gcd(j, n)^k);
%o (PARI) T(n, k) = sumdiv(n, d, eulerphi(n/d)*d^k);
%o (PARI) T(n, k) = sumdiv(n, d, moebius(n/d)*d*sigma(d, k-1));
%Y Columns k=1..7 give A018804, A069097, A343497, A343498, A343499, A343508, A343509.
%Y T(n-2,n) gives A342432.
%Y T(n-1,n) gives A342433.
%Y T(n,n) gives A332517.
%Y T(n,n+1) gives A321294.
%Y Cf. A008292, A343516.
%K nonn,tabl,easy
%O 1,3
%A _Seiichi Manyama_, Apr 17 2021