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A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)). 15

%I #28 Oct 25 2018 15:56:38

%S 1,1,1,1,1,2,1,1,4,1,1,1,10,1,2,1,1,28,1,6,2,1,1,82,1,26,4,2,1,1,244,

%T 1,126,10,8,1,1,1,730,1,626,28,50,1,3,1,1,2188,1,3126,82,344,1,13,2,1,

%U 1,6562,1,15626,244,2402,1,91,6,2,1,1,19684,1,78126,730,16808,1,757,26,12,2

%N Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

%C A(n,k) is the sum of k-th powers of odd divisors of n.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

%F G.f. of column k: Sum_{j>=1} (2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

%e Square array begins:

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

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

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

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

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

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

%t Table[Function[k, SeriesCoefficient[Sum[(2 i - 1)^k x^(2 i - 1)/(1 - x^(2 i - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 1, j}] // Flatten

%Y Columns k=0-5 give: A001227, A000593, A050999, A051000, A051001, A051002.

%Y Cf. A109974, A279394, A292919.

%K nonn,tabl

%O 1,6

%A _Ilya Gutkovskiy_, May 14 2017

%E Offset changed by _Ilya Gutkovskiy_, Oct 25 2018

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)