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A322083 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+d)*d^k.

%I A322083 #22 Jan 07 2024 15:52:07
%S A322083 1,1,-2,1,-3,2,1,-5,4,-1,1,-9,10,-3,2,1,-17,28,-13,6,-4,1,-33,82,-57,
%T A322083 26,-12,2,1,-65,244,-241,126,-50,8,0,1,-129,730,-993,626,-252,50,-3,3,
%U A322083 1,-257,2188,-4033,3126,-1394,344,-45,13,-4,1,-513,6562,-16257,15626,-8052,2402,-441,91,-18,2
%N A322083 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+d)*d^k.
%C A322083 For each k, the k-th column sequence (T(n,k))(n>=1) is a multiplicative function of n, equal to (-1)^(n+1)*(Id_k * 1) in the notation of the Bala link. - _Peter Bala_, Mar 19 2022
%H A322083 Andrew Howroyd, <a href="/A322083/b322083.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals)
%H A322083 Peter Bala, <a href="/A067856/a067856_1.pdf">A signed Dirichlet product of arithmetical functions</a>
%H A322083 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>
%F A322083 G.f. of column k: Sum_{j>=1} (-1)^(j+1)*j^k*x^j/(1 + x^j).
%e A322083 Square array begins:
%e A322083    1,   1,   1,    1,     1,     1,  ...
%e A322083   -2,  -3,  -5,   -9,   -17,   -33,  ...
%e A322083    2,   4,  10,   28,    82,   244,  ...
%e A322083   -1,  -3, -13,  -57,  -241,  -993,  ...
%e A322083    2,   6,  26,  126,   626,  3126,  ...
%e A322083   -4, -12, -50, -252, -1394, -8052,  ...
%t A322083 Table[Function[k, Sum[(-1)^(n/d+d) d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%t A322083 Table[Function[k, SeriesCoefficient[Sum[(-1)^(j + 1) j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%t A322083 f[p_, e_, k_] := If[k == 0, e + 1, (p^(k*e + k) - 1)/(p^k - 1)]; f[2, e_, k_] := If[k == 0, e - 3, -((2^(k - 1) - 1)*2^(k*e + 1) + 2^(k + 1) - 1)/(2^k - 1)]; T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k, k], {n, 1, 11}, {k, n - 1, 0, -1}] // Flatten (* _Amiram Eldar_, Nov 22 2022 *)
%o A322083 (PARI) T(n,k)={sumdiv(n, d, (-1)^(n/d+d)*d^k)}
%o A322083 for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 26 2018
%Y A322083 Columns k=0..12 give A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809.
%Y A322083 Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322084.
%K A322083 sign,tabl
%O A322083 1,3
%A A322083 _Ilya Gutkovskiy_, Nov 26 2018