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

%I A185152 #44 Feb 16 2025 08:33:13
%S A185152 1,6,12,12,30,72,56,24,117,180,132,144,182,336,360,48,306,702,380,360,
%T A185152 672,792,552,288,775,1092,1080,672,870,2160,992,96,1584,1836,1680,
%U A185152 1404,1406,2280,2184,720,1722,4032,1892,1584,3510,3312,2256,576,2793,4650
%N A185152 Expansion of (q/2) * phi(q)^3 (d/dq) phi(q) in powers of q.
%C A185152 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A185152 Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
%H A185152 G. C. Greubel, <a href="/A185152/b185152.txt">Table of n, a(n) for n = 1..1000</a>
%H A185152 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A185152 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A185152 Expansion of (-1/8) * theta_3(0,q)^3 * theta_3(0,q)'' in powers of nome q.
%F A185152 Expansion of (-1/24) * q * (d/dq) (P(q) - 4 * P(q^4)) where P() is a Ramanujan Eisenstein series.
%F A185152 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.
%F A185152 Multiplicative with a(2^e) = 3 * 2^e if e>0, a(p^e) = p^e * (p^(e+1) - 1) / (p - 1) otherwise.
%F A185152 G.f.: Sum_{k>0} k^2 * x^k / (1 + (-x)^k)^2.
%F A185152 G.f.: Sum_{k>0} k^2 * x^k / (1 - x^k)^2 * (mod(k, 4) > 0).
%F A185152 a(n) = n * Sum of divisors of n that are not divisible by 4 = n * A046897(n).
%F A185152 a(n) = - a(-n). for all n in Z. Convolution of A000118 and A186690.
%F A185152 From _Amiram Eldar_, Sep 12 2023: (Start)
%F A185152 Dirichlet g.f.: (1 - 1/2^(2*s-4)) * zeta(s-2) * zeta(s-1).
%F A185152 Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/24 = 0.41123... (A222171) . (End)
%e A185152 G.f. = x + 6*x^2 + 12*x^3 + 28*x^4 + 30*x^5 + 72*x^6 + 56*x^7 + 24*x^8 + ...
%t A185152 a[ n_] := If[ n == 0, 0, n Sum[ d Sign@Mod[d, 4], {d, Divisors@n}]]; (* _Michael Somos_, Jun 20 2015 *)
%t A185152 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 *)
%t A185152 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 *)
%o A185152 (PARI) {a(n) = if( n==0, 0, n * sumdiv( n, d, if( d%4, d)))};
%Y A185152 Cf. A000027, A000118, A000290, A046897, A186690, A222171.
%Y A185152 Cf. A000122, A000700, A004009, A006352, A010054, A013973, A121373.
%K A185152 nonn,easy,mult
%O A185152 1,2
%A A185152 _Michael Somos_, Jan 23 2012