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Expansion of q*(phi(-q)psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
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%I #24 Sep 12 2023 02:23:58

%S 1,-4,4,0,6,-16,8,0,13,-24,12,0,14,-32,24,0,18,-52,20,0,32,-48,24,0,

%T 31,-56,40,0,30,-96,32,0,48,-72,48,0,38,-80,56,0,42,-128,44,0,78,-96,

%U 48,0,57,-124,72,0,54,-160,72,0,80,-120,60,0,62,-128,104,0,84,-192,68,0,96,-192,72,0,74,-152

%N Expansion of q*(phi(-q)psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A121455/b121455.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael D. Hirschhorn and James A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Sellers/sellers32.html">A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.6.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F Expansion of (eta(q)eta(q^8))^4/(eta(q^2)eta(q^4))^2 in powers of q.

%F Euler transform of period 8 sequence [ -4, -2, -4, 0, -4, -2, -4, -4, ...].

%F Multiplicative with a(2)=-4, a(2^e)=0 if e>1, a(p^e)=(p^(e+1)-1)/(p-1) if p>2.

%F a(4n)=0. a(4n+2)=-4*sigma(2n+1). a(2n+1)=sigma(2n+1).

%F G.f. is Fourier series of a weight 2 level 8 cusp form. f(-1/ (8 t)) = -8 t^2 f(t) where q = exp(2 Pi i t).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w)= ( v* (v+2*w)* (u+2*v))^2 -16* (u*w)^3 -u*v*w* (u +2*v +4*w) *(u^2 +16*v^2 +16*w^2 +10*u*v +28*u*w +40*v*w).

%F Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/2^(s-1)) * (1 - 1/2^s) * zeta(s-1) * zeta(s). - _Amiram Eldar_, Sep 12 2023

%e q - 4*q^2 + 4*q^3 + 6*q^5 - 16*q^6 + 8*q^7 + 13*q^9 - 24*q^10 + 12*q^11 + ...

%t a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, q^2] *EllipticTheta[3, 0, -q])^2/4, {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Jan 04 2018 *)

%o (PARI) {a(n)=if(n<1, 0, if(n%2, sigma(n), if(n/2%2, -4*sigma(n/2), 0)))}

%o (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^8+A))^4/(eta(x^2+A)*eta(x^4+A))^2, n))}

%Y Cf. A000122, A000700, A010054, A121373.

%K sign,easy,mult

%O 1,2

%A _Michael Somos_, Jul 30 2006, May 28 2007