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A343510 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{j=1..n} gcd(j, n)^k.

%I A343510 #47 Feb 16 2025 08:34:02
%S A343510 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 A343510 55,13,1,129,731,1058,629,261,55,20,1,257,2189,4162,3129,1411,349,92,
%U A343510 21,1,513,6563,16514,15629,8085,2407,596,105,27,1,1025,19685,65794,78129,47515,16813,4388,789,145,21
%N A343510 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 A343510 Seiichi Manyama, <a href="/A343510/b343510.txt">Antidiagonals n = 1..140, flattened</a>
%H A343510 Peter Bala, <a href="/A368743/a368743.pdf">GCD sum theorems. Two Multivariable Cesaro Type Identities</a>.
%H A343510 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a> and <a href="https://mathworld.wolfram.com/EulersNumberTriangle.html">Euler's Number Triangle</a>
%F A343510 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 A343510 T(n,k) = Sum_{d|n} phi(n/d) * d^k.
%F A343510 T(n,k) = Sum_{d|n} mu(n/d) * d * sigma_{k-1}(d).
%F A343510 Dirichlet g.f. of column k: zeta(s-1) * zeta(s-k) / zeta(s). - _Ilya Gutkovskiy_, Apr 18 2021
%F A343510 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 A343510 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 A343510 G.f. of column 3: Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^4.
%e A343510 Square array begins:
%e A343510    1,  1,   1,    1,     1,      1,      1, ...
%e A343510    3,  5,   9,   17,    33,     65,    129, ...
%e A343510    5, 11,  29,   83,   245,    731,   2189, ...
%e A343510    8, 22,  74,  274,  1058,   4162,  16514, ...
%e A343510    9, 29, 129,  629,  3129,  15629,  78129, ...
%e A343510   15, 55, 261, 1411,  8085,  47515, 282381, ...
%e A343510   13, 55, 349, 2407, 16813, 117655, 823549, ...
%t A343510 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 A343510 (PARI) T(n, k) = sum(j=1, n, gcd(j, n)^k);
%o A343510 (PARI) T(n, k) = sumdiv(n, d, eulerphi(n/d)*d^k);
%o A343510 (PARI) T(n, k) = sumdiv(n, d, moebius(n/d)*d*sigma(d, k-1));
%Y A343510 Columns k=1..7 give A018804, A069097, A343497, A343498, A343499, A343508, A343509.
%Y A343510 T(n-2,n) gives A342432.
%Y A343510 T(n-1,n) gives A342433.
%Y A343510 T(n,n) gives A332517.
%Y A343510 T(n,n+1) gives A321294.
%Y A343510 Cf. A008292, A343516.
%K A343510 nonn,tabl,easy
%O A343510 1,3
%A A343510 _Seiichi Manyama_, Apr 17 2021