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A345229 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).

%I A345229 #34 Sep 13 2024 11:59:28
%S A345229 1,1,3,1,4,6,1,5,9,10,1,6,13,17,15,1,7,18,28,26,21,1,8,24,44,47,41,28,
%T A345229 1,9,31,66,83,82,54,36,1,10,39,95,140,159,116,74,45,1,11,48,132,225,
%U A345229 293,249,172,95,55,1,12,58,178,346,512,509,401,235,122,66,1,13,69,234,512,852,980,888,592,321,143,78
%N A345229 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 A345229 Seiichi Manyama, <a href="/A345229/b345229.txt">Antidiagonals n = 1..140, flattened</a>
%F A345229 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^k.
%F A345229 T(n,k) = Sum_{j=1..n} Sum_{d|j} phi(j/d) * binomial(d+k-2, k-1).
%F A345229 T(n,k) = Sum_{j=1..n} phi(j) * binomial(floor(n/j)+k-1,k). - _Seiichi Manyama_, Sep 13 2024
%e A345229 G.f. of column 3: (1/(1 - x)) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^3.
%e A345229 Square array begins:
%e A345229    1,  1,   1,   1,   1,   1,    1, ...
%e A345229    3,  4,   5,   6,   7,   8,    9, ...
%e A345229    6,  9,  13,  18,  24,  31,   39, ...
%e A345229   10, 17,  28,  44,  66,  95,  132, ...
%e A345229   15, 26,  47,  83, 140, 225,  346, ...
%e A345229   21, 41,  82, 159, 293, 512,  852, ...
%e A345229   28, 54, 116, 249, 509, 980, 1782, ...
%p A345229 T:= (n, k)-> coeff(series((1/(1-x))* add(numtheory[phi](j)
%p A345229              *x^j/(1-x^j)^k, j=1..n), x, n+1), x, n):
%p A345229 seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # _Alois P. Heinz_, Jun 11 2021
%t A345229 T[n_, k_] := Sum[DivisorSum[j, EulerPhi[j/#] * Binomial[k + # - 2, k - 1] &], {j, 1, n}];  Table[T[k, n - k + 1], {n, 1, 12}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jun 11 2021 *)
%o A345229 (PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, eulerphi(j/d)*binomial(d+k-2, k-1)));
%o A345229 (PARI) T(n, k) = sum(j=1, n, eulerphi(j)*binomial(n\j+k-1, k)); \\ _Seiichi Manyama_, Sep 13 2024
%Y A345229 Columns k=1..4 give A000217, A272718, A344521, A344992.
%Y A345229 Main diagonal gives A345230.
%Y A345229 Cf. A343516, A344479.
%K A345229 nonn,tabl
%O A345229 1,3
%A A345229 _Seiichi Manyama_, Jun 11 2021