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%I #23 Apr 14 2021 22:24:19
%S 0,0,0,1,3,4,10,15,24,33,45,65,77,102,143,155,180,268,255,315,434,435,
%T 462,695,593,735,960,918,945,1437,1160,1395,1825,1692,1668,2549,1995,
%U 2385,3073,2775,2730,4190,3157,3747,4739,4290,4140,6355,4686,5523,7044
%N Convolution of level 3 of the divisor function.
%C Named W3(n) by S. Alaca and K. S. Williams.
%H Robert Israel, <a href="/A218277/b218277.txt">Table of n, a(n) for n = 1..10000</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>, arXiv:math/0510429 [math.NT], 2005-2006; International Journal of Number Theory 3, 2 (2007), Pages 231-261.
%F a(n) = Sum_{m<3n} sigma(m)*sigma(n-3*m).
%F a(n) = sigma3(n)/24 - n*sigma(n)/12 + sigma(n)/24 + 3*sigma3(n/3)/8 - n*sigma(n/3)/4 + sigma(n/3)/24.
%F a(n) = (1/72)*(31*sigma_3(n) - sigma_3(3*n) + 7*sigma(n) - sigma(3*n) - 30*n*sigma(n) + 6*n*sigma(3*n)). - _Ridouane Oudra_, Mar 21 2021
%p f:= n -> add(numtheory:-sigma(m)*numtheory:-sigma(n-3*m),m=1..floor((n-1)/3)):
%p map(f, [$1..50]); # _Robert Israel_, Jun 28 2018
%p with(numtheory): seq((1/72)*(31*sigma[3](n) - sigma[3](3*n) + 7*sigma(n) - sigma(3*n) - 30*n*sigma(n) + 6*n*sigma(3*n)), n=1..50); # _Ridouane Oudra_, Mar 21 2021
%t a[n_] := Sum[DivisorSigma[1, m] DivisorSigma[1, n-3m], {m, 1, (n-1)/3}];
%t Array[a, 50] (* _Jean-François Alcover_, Sep 19 2018 *)
%o (PARI) lista(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/3), sigma(m)*sigma(i-3*m)); print1(s , ", "););}
%o (PARI) lista(n) = {for (i=1, n, v = sigma(i,3)/24 - i*sigma(i)/12 + sigma(i)/24;if (i%3 == 0, v += 3*sigma(i/3,3)/8 - i*sigma(i/3)/4 + sigma(i/3)/24); print1(v , ", "););}
%Y Cf. A000385, A218276, A218278.
%K nonn
%O 1,5
%A _Michel Marcus_, Oct 25 2012
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