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Expansion of q * f(q^5)^5 / f(q) in powers of q where f() is a Ramanujan theta function.
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%I #21 Nov 23 2023 08:01:03

%S 1,-1,2,-3,5,-2,6,-5,7,-5,12,-6,12,-6,10,-11,16,-7,20,-15,12,-12,22,

%T -10,25,-12,20,-18,30,-10,32,-21,24,-16,30,-21,36,-20,24,-25,42,-12,

%U 42,-36,35,-22,46,-22,43,-25,32,-36,52,-20,60,-30,40,-30,60,-30,62

%N Expansion of q * f(q^5)^5 / f(q) in powers of q where f() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A138512/b138512.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%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^4) * eta(q^10)^15 / (eta(q^2)^3 * eta(q^5)^5 * eta(q^20)^5) in powers of q.

%F Euler transform of period 20 sequence [ -1, 2, -1, 1, 4, 2, -1, 1, -1, -8, -1, 1, -1, 2, 4, 1, -1, 2, -1, -4, ...].

%F a(n) is multiplicative with a(2^e) = -(2^(e+1) - (-1)^(e+1)) / 3 if e>0, a(5^e) = 5^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), a(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 2, 3 (mod 5).

%F a(n) = -(-1)^n * A053723(n-1).

%F From _Michael Somos_, Sep 25 2015: (Start)

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (16/5)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138506.

%F a(2*n + 1) = A134080(n). (End)

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = A328717 / 8 = 0.0882764... . - _Amiram Eldar_, Nov 23 2023

%e G.f. = q - q^2 + 2*q^3 - 3*q^4 + 5*q^5 - 2*q^6 + 6*q^7 - 5*q^8 + 7*q^9 - 5*q^10 + ...

%t a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^5]^5 / QPochhammer[ -q], {q, 0, n}]; (* _Michael Somos_, Sep 25 2015 *)

%t a[ n_] := If[ n < 1, 0, -(-1)^n Sum[ n/d KroneckerSymbol[ 5, d], { d, Divisors @ n}]]; (* _Michael Somos_, Sep 25 2015 *)

%o (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, d * kronecker(5, n/d)))};

%o (PARI) {a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(2^(e+1) - (-1)^(e+1)) / 3, f = kronecker(5, p); (p^(e+1) - f^(e+1)) / (p - f) ))) };

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A)^15 / (eta(x^2 + A)^3 * eta(x^5 + A)^5 * eta(x^20 + A)^5), n))};

%Y Cf. A053723, A134080, A138506, A328717.

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

%K sign,easy,mult

%O 1,3

%A _Michael Somos_, Mar 21 2008