A282097 Coefficients in q-expansion of (3*E_2*E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 12, 36, 112, 150, 432, 392, 960, 1053, 1800, 1452, 4032, 2366, 4704, 5400, 7936, 5202, 12636, 7220, 16800, 14112, 17424, 12696, 34560, 19375, 28392, 29160, 43904, 25230, 64800, 30752, 64512, 52272, 62424, 58800, 117936, 52022, 86640, 85176, 144000, 70602

Offset

0,3

Comments

Multiplicative because A000203 is. - Andrew Howroyd, Jul 25 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = (3*A282019(n) - 2*A013973(n) - A282018(n))/1728.

G.f.: phi_{3, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

a(n) = n^2*A000203(n) for n > 0. - Seiichi Manyama, Feb 19 2017

G.f.: Sum_{k>=1} k^3*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, May 02 2018

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^(2*e) * (p^(e+1)-1)/(p-1).

Dirichlet g.f.: zeta(s-2)*zeta(s-3).

Sum_{k=1..n} a(k) ~ (Pi^2/24) * n^4. (End)

From Peter Bala, Jan 22 2024:

a(n) = Sum_{1 <= i, j, k <= n} sigma_2( gcd(i, j, k, n) ).

a(n) = Sum_{1 <= i, j <= n} sigma_3( gcd(i, j, n) ).

a(n) = Sum_{d divides n} sigma_2(d) * J_3(n/d) = Sum_{d divides n} sigma_3(d) * J_2(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n). (End)

Example

a(6) = 1^3*6^2 + 2^3*3^2 + 3^3*2^2 + 6^3*1^2 = 432.

Mathematica

a[0]=0; a[n_]:=(n^2)*DivisorSigma[1, n]; Table[a[n], {n, 0, 41}] (* Indranil Ghosh, Feb 21 2017 *)
terms = 42; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(3*Ei[2]*Ei[4] - 2*Ei[6] - Ei[2]^3)/1728 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

Prog

(PARI) a(n) = if (n==0, 0, n^2*sigma(n)); \\ Michel Marcus, Feb 21 2017
(Magma) [0] cat [n^2*DivisorSigma(1, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

Crossrefs

Cf. this sequence (phi_{3, 2}), A282099 (phi_{5, 2}).

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4).

Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), this sequence (n^2*sigma(n)), A282211 (n^3*sigma(n)).

Cf. A222171.

Sequence in context: A172212 A060621 A058880 * A055551 A355386 A073403

Adjacent sequences: A282094 A282095 A282096 * A282098 A282099 A282100

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Feb 06 2017

Status

approved