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A185152
Expansion of (q/2) * phi(q)^3 (d/dq) phi(q) in powers of q.
4
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
1,2
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).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
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)
EXAMPLE
G.f. = x + 6*x^2 + 12*x^3 + 28*x^4 + 30*x^5 + 72*x^6 + 56*x^7 + 24*x^8 + ...
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 *)
PROG
(PARI) {a(n) = if( n==0, 0, n * sumdiv( n, d, if( d%4, d)))};
KEYWORD
nonn,easy,mult
AUTHOR
Michael Somos, Jan 23 2012
STATUS
approved