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