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A322081 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+1)*d^k.

%I A322081 #13 Jan 07 2024 15:52:19
%S A322081 1,1,0,1,1,2,1,3,4,-1,1,7,10,1,2,1,15,28,11,6,0,1,31,82,55,26,4,2,1,
%T A322081 63,244,239,126,30,8,-2,1,127,730,991,626,196,50,1,3,1,255,2188,4031,
%U A322081 3126,1230,344,43,13,0,1,511,6562,16255,15626,7564,2402,439,91,6,2,1,1023,19684,65279,78126,45990,16808,3823,757,78,12,-2
%N A322081 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+1)*d^k.
%H A322081 Andrew Howroyd, <a href="/A322081/b322081.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals)
%H A322081 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>
%F A322081 G.f. of column k: Sum_{j>=1} j^k*x^j/(1 + x^j).
%e A322081 Square array begins:
%e A322081    1,  1,   1,    1,     1,     1,  ...
%e A322081    0,  1,   3,    7,    15,    31,  ...
%e A322081    2,  4,  10,   28,    82,   244,  ...
%e A322081   -1,  1,  11,   55,   239,   991,  ...
%e A322081    2,  6,  26,  126,   626,  3126,  ...
%e A322081    0,  4,  30,  196,  1230,  7564,  ...
%t A322081 Table[Function[k, Sum[(-1)^(n/d + 1) d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
%t A322081 Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
%o A322081 (PARI) T(n,k)={sumdiv(n, d, (-1)^(n/d+1)*d^k)}
%o A322081 for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 26 2018
%Y A322081 Columns k=0..12 give A048272, A000593, A078306, A078307, A284900, A284926, A284927, A321552,  A321553, A321554, A321555, A321556, A321557.
%Y A322081 Cf. A109974, A279394, A279396, A285425, A321438 (diagonal), A322082, A322083, A322084.
%K A322081 sign,tabl
%O A322081 1,6
%A A322081 _Ilya Gutkovskiy_, Nov 26 2018