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A320973 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = n^k * Product_{p|n, p prime} (1 + 1/p^k).

%I A320973 #7 Dec 09 2018 13:11:56
%S A320973 1,1,2,1,3,2,1,5,4,2,1,9,10,6,2,1,17,28,20,6,4,1,33,82,72,26,12,2,1,
%T A320973 65,244,272,126,50,8,2,1,129,730,1056,626,252,50,12,2,1,257,2188,4160,
%U A320973 3126,1394,344,80,12,4,1,513,6562,16512,15626,8052,2402,576,90,18,2
%N A320973 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = n^k * Product_{p|n, p prime} (1 + 1/p^k).
%H A320973 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dedekind_psi_function">Dedekind psi function</a>
%F A320973 G.f. of column k: Sum_{j>=1} mu(j)^2*PolyLog(-k,x^j), where PolyLog() is the polylogarithm function.
%F A320973 A(n,k) = Sum_{d|n} mu(n/d)^2*d^k.
%e A320973 Square array begins:
%e A320973   1,   1,   1,    1,     1,     1,  ...
%e A320973   2,   3,   5,    9,    17,    33,  ...
%e A320973   2,   4,  10,   28,    82,   244,  ...
%e A320973   2,   6,  20,   72,   272,  1056,  ...
%e A320973   2,   6,  26,  126,   626,  3126,  ...
%e A320973   4,  12,  50,  252,  1394,  8052,  ...
%t A320973 Table[Function[k, n^k Product[1 + Boole[PrimeQ[d]]/d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%t A320973 Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j]^2 PolyLog[-k, x^j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%t A320973 Table[Function[k, Sum[MoebiusMu[n/d]^2 d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%Y A320973 Columns k=0..4 give A034444, A001615, A065958, A065959, A065960.
%Y A320973 Cf. A008683, A059379, A059380, A320974 (diagonal).
%K A320973 nonn,tabl
%O A320973 1,3
%A A320973 _Ilya Gutkovskiy_, Oct 25 2018