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%I #19 Mar 19 2021 18:41:59
%S 0,0,1,3,7,16,22,45,49,100,95,178,161,304,250,465,372,676,525,952,720,
%T 1280,946,1702,1217,2156,1570,2764,1925,3376,2360,4185,2912,4944,3404,
%U 6121,4047,6960,4858,8344,5530,9600,6391,11246,7513,12496,8372,14926,9486
%N Convolution of level 2 of the divisor function.
%C Belongs to the family of convolution sums: Sum_{m < n*N} sigma(n)*sigma(n - N*m).
%C Named W2(n) by S. Alaca and K. S. Williams.
%C The convolution sum: Sum_{m < n} sigma(n)*sigma(n - m) = W1(n) is A000385(n+1).
%H G. C. Greubel, <a href="/A218276/b218276.txt">Table of n, a(n) for n = 1..1000</a>
%H S. Alaca and K. S. Williams, <a href="http://dx.doi.org/10.1016/j.jnt.2006.10.004">Evaluation of the convolution sums ...</a>, Journal of Number Theory, Volume 124, Issue 2, June 2007, Pages 491-510.
%H E. Royer, <a href="http://arxiv.org/abs/math/0510429">Evaluating convolutions of divisor sums with quasimodular forms</a>, International Journal of Number Theory 3, 2 (2007), Pages 231-261.
%F a(n) = Sum_{m < 2*n} sigma(n)*sigma(n - 2*m).
%F a(n) = sigma_3(n)/12 + sigma_3(n/2)/3 - n*sigma(n)/8 - n*sigma(n/2)/4 + sigma(n)/24 + sigma(n/2)/24.
%F a(n) = (1/48)*(22*sigma_3(n) - 2*sigma_3(2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)). - _Ridouane Oudra_, Feb 23 2021
%p with(numtheory): seq((1/48)*(22*sigma[3](n) - 2*sigma[3](2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)),n=1..60); # _Ridouane Oudra_, Feb 23 2021
%t Table[Sum[DivisorSigma[1, k]*DivisorSigma[1, n - 2*k], {k, 1, Floor[(n - 1)/2]}], {n, 1, 50}] (* _G. C. Greubel_, Dec 24 2016 *)
%o (PARI) lista(nn) = {for (i=1, nn, s = sum(m=1, floor((i-1)/2), sigma(m)*sigma(i-2*m)); print1(s , ", "););}
%o (PARI) lista(nn) = {for (i=1, nn, v = sigma(i,3)/12 - i*sigma(i)/8 + sigma(i)/24;if (i%2 == 0, v += sigma(i/2,3)/3 - i*sigma(i/2)/4 + sigma(i/2)/24); print1(v , ", "););}
%Y Cf. A000385, A218277, A218278.
%K nonn
%O 1,4
%A _Michel Marcus_, Oct 25 2012
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