OFFSET
1,6
COMMENTS
Named W4(n) by S. Alaca and K. S. Williams.
LINKS
S. Alaca and K. S. Williams, Evaluation of the convolution sums ..., Journal of Number Theory, Volume 124, Issue 2, June 2007, Pages 491-510.
E. Royer, Evaluating convolutions of divisor sums with quasimodular forms, arXiv:math/0510429 [math.NT], 2005-2006; International Journal of Number Theory 3, 2 (2007) p. 231-261.
FORMULA
a(n) = Sum_{m<4n} sigma(n)*sigma(n-4*m).
a(n) = sigma_3(n)/48 - n*sigma(n)/16 + sigma(n)/24 + sigma_3(n/4)/3 - n*sigma(n/4)/4 + sigma(n/4)/24 + sigma_3(n/2)/16.
a(n) = (1/48)*(sigma_3(n) + 2*sigma(n) - 3*n*sigma(n)) + (1/768)*((1 + (-1)^n))*(173*sigma_3(n) - 21*sigma_3(2*n) + 28*sigma(n) - 12*sigma(2*n) - 168*n*sigma(n) + 72*n*sigma(2*n)). - Ridouane Oudra, Nov 23 2022
MAPLE
with(numtheory): seq(add(sigma(k)*sigma(n-4*k), k=1..floor(n/4)), n=1..70); # Ridouane Oudra, Nov 23 2022
PROG
(PARI) a(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/4), sigma(m)*sigma(i-4*m)); print1(s , ", "); ); }
(PARI) a(n) = {for (i=1, n, v = sigma(i, 3)/48 - i*sigma(i)/16 + sigma(i)/24; if (i%4 == 0, v += sigma(i/4, 3)/3 - i*sigma(i/4)/4 + sigma(i/4)/24); if (i%2 == 0, v += sigma(i/2, 3)/16); print1(v , ", "); ); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 25 2012
STATUS
approved