A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 18, 84, 292, 630, 1512, 2408, 4680, 6813, 11340, 14652, 24528, 28574, 43344, 52920, 74896, 83538, 122634, 130340, 183960, 202272, 263736, 279864, 393120, 393775, 514332, 551880, 703136, 707310, 952560, 923552, 1198368, 1230768, 1503684, 1517040, 1989396, 1874198, 2346120, 2400216, 2948400

Offset

0,3

Comments

The q-expansion of the square of this expression is given in A281371.

Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

Links

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

Formula

a(n) = A145094(n)/240 for n > 0. - Seiichi Manyama, Feb 04 2017

G.f.: phi_{4, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - Seiichi Manyama, Feb 04 2017

a(n) = n*A001158(n) for n > 0. - Seiichi Manyama, Feb 18 2017

G.f.: x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017

Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / 450. - Vaclav Kotesovec, May 09 2022

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^e * (p^(3*e+3)-1)/(p^3-1).

Dirichlet g.f.: zeta(s-1)*zeta(s-4). (End)

a(n) = Sum_{k = 1..n} sigma_4( gcd(k, n) ) = Sum_{d divides n} sigma_4(d) * phi(n/d). - Peter Bala, Jan 19 2024

a(n) = Sum_{1 <= i, j, k, l <= n} sigma_1( gcd(i, j, k, l, n) ) = Sum_{d divides n} sigma_1(d) * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 22 2024

Maple

with(gfun):
with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
t1:=series((e2*e4-e6)/720, q, M+1);
seriestolist(t1);
# alternative program
seq(add(sigma[4](d)*phi(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 20 2024

Mathematica

Table[If[n==0, 0, n * DivisorSigma[3, n]], {n, 0, 40}] (* Indranil Ghosh, Mar 11 2017 *)
terms = 41; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(Ei[2] Ei[4] - Ei[6])/720 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

Prog

(PARI) for(n=0, 40, print1(if(n==0, 0, n * sigma(n, 3)), ", ")) \\ Indranil Ghosh, Mar 11 2017
(Magma) [0] cat [n*DivisorSigma(3, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

Crossrefs

Cf. A001158, A003840, A006352, A004009, A013973, A064987, A145094, A281371, A328259.

Sequence in context: A039358 A043961 A067984 * A124935 A126405 A250101

Adjacent sequences: A281369 A281370 A281371 * A281373 A281374 A281375

Keyword

nonn,easy,mult

Author

N. J. A. Sloane, Feb 04 2017

Status

approved