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A343516 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, n).

%I A343516 #25 Apr 20 2021 10:09:18
%S A343516 1,1,3,1,4,5,1,5,8,8,1,6,12,15,9,1,7,17,26,19,15,1,8,23,42,39,35,13,1,
%T A343516 9,30,64,74,76,34,20,1,10,38,93,130,153,90,56,21,1,11,47,130,214,287,
%U A343516 216,152,63,27,1,12,57,176,334,506,468,379,191,86,21
%N A343516 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, n).
%H A343516 Seiichi Manyama, <a href="/A343516/b343516.txt">Antidiagonals n = 1..140, flattened</a>
%F A343516 G.f. of column k: Sum_{j>=1} phi(j) * x^j/(1 - x^j)^(k+1).
%F A343516 T(n,k) = Sum_{d|n} phi(n/d) * binomial(d+k-1, k).
%e A343516 T(4,2) = gcd(1,1,4) + gcd(1,2,4) + gcd(2,2,4) + gcd(1,3,4) + gcd(2,3,4) + gcd(3,3,4) + gcd(1,4,4) + gcd(2,4,4) + gcd(3,4,4) + gcd(4,4,4) = 1 + 1 + 2 + 1 + 1 + 1 + 1 + 2 + 1 + 4 = 15.
%e A343516 Square array begins:
%e A343516    1,  1,  1,   1,   1,   1,    1, ...
%e A343516    3,  4,  5,   6,   7,   8,    9, ...
%e A343516    5,  8, 12,  17,  23,  30,   38, ...
%e A343516    8, 15, 26,  42,  64,  93,  130, ...
%e A343516    9, 19, 39,  74, 130, 214,  334, ...
%e A343516   15, 35, 76, 153, 287, 506,  846, ...
%e A343516   13, 34, 90, 216, 468, 930, 1722, ...
%t A343516 T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * Binomial[k + # - 1, k] &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Apr 18 2021 *)
%o A343516 (PARI) T(n, k) = sumdiv(n, d, eulerphi(n/d)*binomial(d+k-1, k));
%Y A343516 Columns k=1..7 give A018804, A309322, A309323, A343518, A343519, A343520, A343521.
%Y A343516 Main diagonal gives A343517.
%Y A343516 T(n,n-1) gives A343553.
%Y A343516 Cf. A343510.
%K A343516 nonn,tabl
%O A343516 1,3
%A A343516 _Seiichi Manyama_, Apr 17 2021