login
Expansion of q^(-1/2) * (eta(q)^4 * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.
2

%I #19 Feb 16 2025 08:33:07

%S 1,-8,26,-48,73,-120,170,-208,290,-360,384,-528,651,-656,842,-960,960,

%T -1248,1370,-1360,1682,-1848,1898,-2208,2353,-2320,2810,-3120,2880,

%U -3480,3722,-3504,4420,-4488,4224,-5040,5330,-5208,5760,-6240,5905,-6888,7540,-6736,7922,-8160,7680

%N Expansion of q^(-1/2) * (eta(q)^4 * eta(q^4)^2 / 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="/A138502/b138502.txt">Table of n, a(n) for n = 0..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 (phi(-q)^2 * psi(q^2))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

%F Euler transform of period 4 sequence [ -8, -2, -8, -6, ...].

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 256 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138501.

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

%F a(n) = (-1)^n * A122854(n) = A002173(2*n + 1).

%e G.f. = 1 - 8*x + 26*x^2 - 48*x^3 + 73*x^4 - 120*x^5 + 170*x^6 - 208*x^7 + ...

%e g.f. = q - 8*q^3 + 26*q^5 - 48*q^7 + 73*q^9 - 120*q^11 + 170*q^13 - 208*q^15 + ...

%t a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, #^2 KroneckerSymbol[ -4, #] &]]; (* _Michael Somos_, Aug 26 2015 *)

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

%t a[ n_] := If[ n < 0, 0, Times @@ (Function[ {p, e}, If[ p < 3, 2 - p, With[{f = (-1)^Quotient[p, 2]}, f ((f p^2)^(e + 1) - 1)/(p^2 - f)]]]) @@@ FactorInteger[2 n + 1]]; (* _Michael Somos_, Aug 26 2015 *)

%o (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, d^2 * kronecker( -4, d)))};

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

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

%Y Cf. A002173, A122854.

%K sign,changed

%O 0,2

%A _Michael Somos_, Mar 20 2008