%I #14 Feb 16 2025 08:33:07

%S 1,-4,8,-16,26,-32,48,-64,73,-104,120,-128,170,-192,208,-256,290,-292,

%T 360,-416,384,-480,528,-512,651,-680,656,-768,842,-832,960,-1024,960,

%U -1160,1248,-1168,1370,-1440,1360,-1664,1682,-1536,1848,-1920,1898,-2112,2208,-2048,2353,-2604

%N Expansion of (eta(q)^2 * eta(q^4)^4 / eta(q^2)^3)^2 in powers of q.

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

%H G. C. Greubel, <a href="/A138501/b138501.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F Euler transform of period 4 sequence [ -4, 2, -4, -6, ...].

%F a(n) is multiplicative with a(2^e) = -(4^e) if e>0, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), a(p^e) = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 3 (mod 4).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138502.

%F G.f.: x * (Product_{k>0} (1 - x^k)^3 * (1 + x^k) * (1 + x^(2*k))^4)^2.

%F a(n) = -(-1)^n * A050470(n).

%e G.f. = q - 4*q^2 + 8*q^3 - 16*q^4 + 26*q^5 - 32*q^6 + 48*q^7 - 64*q^8 + 73*q^9 + ...

%t a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, #^2 Mod[n/#, 2] (-1)^Quotient[n/#, 2] &]]; (* _Michael Somos_, Aug 26 2015 *)

%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q]^2 QPochhammer[ q^4]^4 / QPochhammer[ q^2]^3)^2, {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)

%o (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, d^2 * (n / d % 2) * (-1)^(n / d \ 2)))};

%o (PARI) {a(n) = my(A, p, e, f); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -4^e, f = (-1)^(p\2); ((p^2)^(e+1) - f^(e+1)) / (p^2 - f))))};

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n) ; polcoeff( (eta(x + A)^2 * eta(x^4 + A)^4 / eta(x^2 + A)^3)^2, n))};

%Y Cf. A050470, A138502.

%K sign,mult

%O 1,2

%A _Michael Somos_, Mar 20 2008