A347227 - OEIS

%I #18 Aug 25 2021 07:42:18

%S 1,1,-2,1,-3,-2,1,-5,-4,1,1,-9,-10,2,-2,1,-17,-28,4,-6,4,1,-33,-82,8,

%T -26,12,-2,1,-65,-244,16,-126,50,-8,0,1,-129,-730,32,-626,252,-50,0,1,

%U 1,-257,-2188,64,-3126,1394,-344,0,3,4,1,-513,-6562,128,-15626,8052,-2402,0,9,18,-2

%N Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} mu(d)*mu(n/d)*d^k.

%H Seiichi Manyama, <a href="/A347227/b347227.txt">Antidiagonals n = 1..140, flattened</a>

%F Dirichlet g.f. of column k: 1/(zeta(s)*zeta(s-k)).

%e Square array begins:

%e    1,  1,   1,    1,    1,     1, ...

%e   -2, -3,  -5,   -9,  -17,   -33, ...

%e   -2, -4, -10,  -28,  -82,  -244, ...

%e    1,  2,   4,    8,   16,    32, ...

%e   -2, -6, -26, -126, -626, -3126, ...

%e    4, 12,  50,  252, 1394,  8052, ...

%t T[n_, k_] := DivisorSum[n, MoebiusMu[#] * MoebiusMu[n/#] * #^k &]; Table[T[n - k + 1, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Amiram Eldar_, Aug 24 2021 *)

%o (PARI) T(n, k) = sumdiv(n, d, moebius(d)*moebius(n/d)*d^k);

%Y Columns k=0..5 give A007427, A046692, A053822, A053825, A053826, A178448.

%Y T(n,n) gives A347251.

%K sign,tabl

%O 1,3

%A _Seiichi Manyama_, Aug 24 2021