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A322082 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d odd} d^k.

%I A322082 #15 Jan 07 2024 15:52:11
%S A322082 1,1,1,1,2,2,1,4,4,1,1,8,10,4,2,1,16,28,16,6,2,1,32,82,64,26,8,2,1,64,
%T A322082 244,256,126,40,8,1,1,128,730,1024,626,224,50,8,3,1,256,2188,4096,
%U A322082 3126,1312,344,64,13,2,1,512,6562,16384,15626,7808,2402,512,91,12,2,1,1024,19684,65536,78126,46720,16808,4096,757,104,12,2
%N A322082 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d odd} d^k.
%H A322082 Andrew Howroyd, <a href="/A322082/b322082.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals)
%H A322082 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>
%F A322082 G.f. of column k: Sum_{j>=1} j^k*x^j/(1 - x^(2*j)).
%e A322082 Square array begins:
%e A322082   1,  1,   1,    1,     1,     1,  ...
%e A322082   1,  2,   4,    8,    16,    32,  ...
%e A322082   2,  4,  10,   28,    82,   244,  ...
%e A322082   1,  4,  16,   64,   256,  1024,  ...
%e A322082   2,  6,  26,  126,   626,  3126,  ...
%e A322082   2,  8,  40,  224,  1312,  7808,  ...
%t A322082 Table[Function[k, Sum[Boole[OddQ[n/d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
%t A322082 Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 - x^(2 j)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
%o A322082 (PARI) T(n,k)={sumdiv(n, d, if(n/d%2, d^k))}
%o A322082 for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 26 2018
%Y A322082 Columns k=0..12 give A001227, A002131, A076577, A007331, A285989, A096960, A321817, A096961, A321818, A096962, A321819, A096963, A321820.
%Y A322082 Cf. A109974, A279394, A279396, A285425, A322081, A322083, A322084.
%K A322082 nonn,tabl
%O A322082 1,5
%A A322082 _Ilya Gutkovskiy_, Nov 26 2018