A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

0, 1, 2052, 177156, 4202512, 48828150, 363524112, 1977326792, 8606744640, 31382654013, 100195363800, 285311670732, 744500215872, 1792160394206, 4057474577184, 8650199741400, 17626613022976, 34271896307922, 64397206034676, 116490258898580, 205200886312800

Offset

0,3

Comments

Multiplicative because A013957 is. - Andrew Howroyd, Jul 23 2018

Links

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

Formula

a(n) = n^2*A013957(n) for n > 0.

a(n) = (6*A282549(n) - 5*A282792(n) + 4*A282576(n) - 5*A058550(n))/1728.

Sum_{k=1..n} a(k) ~ zeta(10) * n^12 / 12. - Amiram Eldar, Sep 06 2023

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^(2*e) * (p^(9*e+9)-1)/(p^9-1).

Dirichlet g.f.: zeta(s-2)*zeta(s-11). (End)

Mathematica

Table[If[n>0, n^2 * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)

Prog

(PARI) for(n=0, 20, print1(if(n==0, 0, n^2 * sigma(n, 9)), ", ")) \\ Indranil Ghosh, Mar 12 2017

Crossrefs

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), A282753 (phi_{9, 2}), this sequence (phi_{11, 2}).

Cf. A282549 (E_2*E_4^3), A282792 (E_2^2*E_4*E_6), A282576 (E_2*E_6^2), A058550 (E_4^2*E_6 = E_14).

Cf. A013957 (sigma_9(n)), A282254 (n*sigma_9(n)), this sequence (n^2*sigma_9(n)).

Cf. A013668 (zeta(10)).

Sequence in context: A320277 A322102 A206068 * A168190 A238934 A084550

Adjacent sequences: A280018 A280019 A280020 * A280022 A280023 A280024

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Feb 22 2017

Status

approved