OFFSET
1,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
FORMULA
Coefficients in expansion of Sum_{n >= 1} x^n/(1-x^n)^3.
G.f.: Sum_{n>=1} (n*(n+1)/2) * x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
a(n) = Sum_{d|n} d*(d+1)/2 = (1/2)*(sigma(n) + sigma_2(n)) = (1/2)*(A000203(n) + A001157(n)). - Benoit Cloitre, Apr 08 2002
Row sums of triangle A134839 - Gary W. Adamson, Nov 12 2007
Dirichlet g.f. zeta(s)*(zeta(s-1) + zeta(s-2))/2. - Franklin T. Adams-Watters, Nov 05 2009
L.g.f.: -log(Product_{k>=1} (1 - x^k)^((k+1)/2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 12 2018
Sum_{k=1..n} a(k) ~ Zeta(3) * n^3 / 6. - Vaclav Kotesovec, Nov 06 2018
a(n) = Sum_{i=1..n} i*A135539(n,i). - Ridouane Oudra, Jul 22 2022
MAPLE
with(numtheory):
a:= proc(n) option remember;
add(d*(d+1)/2, d=divisors(n))
end:
seq(a(n), n=1..60); # Alois P. Heinz, Feb 09 2011
MATHEMATICA
a[n_] := (DivisorSigma[1, n] + DivisorSigma[2, n])/2; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 10 2014, after Benoit Cloitre *)
PROG
(PARI) a(n)=if(n<1, 1, sumdiv(n, d, (d^2+d))/2); /* Joerg Arndt, Aug 14 2012 */
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+1, 2)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Christian G. Bower.
STATUS
approved