A007437 Inverse Moebius transform of triangular numbers.

1, 4, 7, 14, 16, 31, 29, 50, 52, 74, 67, 119, 92, 137, 142, 186, 154, 247, 191, 294, 266, 323, 277, 455, 341, 446, 430, 553, 436, 686, 497, 714, 634, 752, 674, 1001, 704, 935, 878, 1150, 862, 1298, 947, 1323, 1222, 1361, 1129, 1767, 1254, 1674, 1486, 1834

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Crossrefs

Cf. A134544, A134545.

Cf. A134839, A301873, A301874.

Cf. A135539.

Programs

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 *)

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

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 triangles A134544 and A134545. - Gary W. Adamson, Oct 31 2007

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

Extensions

More terms from Christian G. Bower.