A185152 Expansion of (q/2) * phi(q)^3 (d/dq) phi(q) in powers of q.

1, 6, 12, 12, 30, 72, 56, 24, 117, 180, 132, 144, 182, 336, 360, 48, 306, 702, 380, 360, 672, 792, 552, 288, 775, 1092, 1080, 672, 870, 2160, 992, 96, 1584, 1836, 1680, 1404, 1406, 2280, 2184, 720, 1722, 4032, 1892, 1584, 3510, 3312, 2256, 576, 2793, 4650

Offset: 1

Author

Michael Somos, Jan 23 2012

Keywords

• nonn
• easy
• mult

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

Examples

			G.f. = x + 6*x^2 + 12*x^3 + 28*x^4 + 30*x^5 + 72*x^6 + 56*x^7 + 24*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A000027, A000118, A000290, A046897, A186690, A222171.

Cf. A000122, A000700, A004009, A006352, A010054, A013973, A121373.

Programs

Mathematica

a[ n_] := If[ n == 0, 0, n Sum[ d Sign@Mod[d, 4], {d, Divisors@n}]]; (* Michael Somos, Jun 20 2015 *)
a[ n_] := Sign[n] With[ {f = Series[ EllipticTheta[ 3, 0, q], {q, 0, Abs@n + 1}]}, SeriesCoefficient[ (q/2) f^3 D[f, q], Abs@n]]; (* Michael Somos, Jun 20 2015 *)
a[ n_] := Sign[n] With[ {f = Series[ EllipticTheta[ 3, x, q], {q, 0, Abs@n + 1}]}, SeriesCoefficient[ (-1/8) f^3 D[f, x, x] /. x -> 0, Abs@n]]; (* Michael Somos, Jun 20 2015 *)

PARI

{a(n) = if( n==0, 0, n * sumdiv( n, d, if( d%4, d)))};

Formula

Expansion of (-1/8) * theta_3(0,q)^3 * theta_3(0,q)'' in powers of nome q.

Expansion of (-1/24) * q * (d/dq) (P(q) - 4 * P(q^4)) where P() is a Ramanujan Eisenstein series.

Expansion of (1/8) * (E(k^2) - (1-k^2) * K(k^2)) * K(k^2)^3 / (Pi/2)^4 in powers of nome q.

Multiplicative with a(2^e) = 3 * 2^e if e>0, a(p^e) = p^e * (p^(e+1) - 1) / (p - 1) otherwise.

G.f.: Sum_{k>0} k^2 * x^k / (1 + (-x)^k)^2.

G.f.: Sum_{k>0} k^2 * x^k / (1 - x^k)^2 * (mod(k, 4) > 0).

a(n) = n * Sum of divisors of n that are not divisible by 4 = n * A046897(n).

a(n) = - a(-n). for all n in Z. Convolution of A000118 and A186690.

From Amiram Eldar, Sep 12 2023: (Start)

Dirichlet g.f.: (1 - 1/2^(2*s-4)) * zeta(s-2) * zeta(s-1).

Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/24 = 0.41123... (A222171) . (End)