A138506 Expansion of f(q)^5 / f(q^5) in powers of q where f() is a Ramanujan theta function.

1, 5, 5, -10, -15, 5, -10, -30, 25, 35, 5, 60, 30, -60, -30, -10, -55, -80, 35, 100, -15, 60, 60, -110, -50, 5, -60, -100, 90, 150, -10, 160, 105, -120, -80, -30, -105, -180, 100, 120, 25, 210, 60, -210, -180, 35, -110, -230, 110, 215, 5, 160, 180, -260, -100

Offset: 0

Author

Michael Somos, Mar 21 2008, Mar 25 2008

Keywords

• sign

Comments

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

Examples

			G.f. = 1 + 5*q + 5*q^2 - 10*q^3 - 15*q^4 + 5*q^5 - 10*q^6 - 30*q^7 + 25*q^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A109064, A138507.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^5 / QPochhammer[ -q^5], {q, 0, n}]; (* Michael Somos, May 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -5 (-1)^n DivisorSum[ n, # KroneckerSymbol[ 5, #] &]]; (* Michael Somos, May 24 2015 *)

PARI

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

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(-x + A)^5 / eta(-x^5 + A), n))};

Formula

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

Euler transform of period 20 sequence [ 5, -10, 5, -5, 4, -10, 5, -5, 5, -8, 5, -5, 5, -10, 4, -5, 5, -10, 5, -4, ...].

a(n) = 5*b(n) where b() is multiplicative with b(2^e) = ((-2)^(n+1) - 1) / 3, b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 2, 3 (mod 5), b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5).

G.f.: Product_{k>0} (1 - (-x)^k)^5 / (1 - (-x)^(5*k)).

a(n) = (-1)^n * A109064(n). a(n) = 5 * A138507(n) unless n=0.

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(3*sqrt(5)) = 1.471273... . - Amiram Eldar, Jan 29 2024