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%I M3309 #72 Sep 15 2022 08:33:48
%S 1,4,7,14,16,31,29,50,52,74,67,119,92,137,142,186,154,247,191,294,266,
%T 323,277,455,341,446,430,553,436,686,497,714,634,752,674,1001,704,935,
%U 878,1150,862,1298,947,1323,1222,1361,1129,1767,1254,1674,1486,1834
%N Inverse Moebius transform of triangular numbers.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A007437/b007437.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F Coefficients in expansion of Sum_{n >= 1} x^n/(1-x^n)^3.
%F G.f.: Sum_{n>=1} (n*(n+1)/2) * x^n/(1-x^n). - _Joerg Arndt_, Jan 30 2011
%F 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
%F Row sums of triangles A134544 and A134545. - _Gary W. Adamson_, Oct 31 2007
%F Row sums of triangle A134839 - _Gary W. Adamson_, Nov 12 2007
%F Dirichlet g.f. zeta(s)*(zeta(s-1) + zeta(s-2))/2. - _Franklin T. Adams-Watters_, Nov 05 2009
%F 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
%F Sum_{k=1..n} a(k) ~ Zeta(3) * n^3 / 6. - _Vaclav Kotesovec_, Nov 06 2018
%F a(n) = Sum_{i=1..n} i*A135539(n,i). - _Ridouane Oudra_, Jul 22 2022
%p with(numtheory):
%p a:= proc(n) option remember;
%p add(d*(d+1)/2, d=divisors(n))
%p end:
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Feb 09 2011
%t a[n_] := (DivisorSigma[1, n] + DivisorSigma[2, n])/2; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Mar 10 2014, after _Benoit Cloitre_ *)
%o (PARI) a(n)=if(n<1,1,sumdiv(n,d,(d^2+d))/2); /* _Joerg Arndt_, Aug 14 2012 */
%o (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
%Y Cf. A134544, A134545.
%Y Cf. A134839, A301873, A301874.
%Y Cf. A135539.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Christian G. Bower_.
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