A122190 Expansion of q^(-1/4) * eta(q^2) * eta(q^5)^3 / (eta(q) * eta(q^10)) in powers of q.

1, 1, 1, 2, 2, 0, 1, 2, 0, 2, 2, 1, 1, 2, 0, 2, 2, 0, 2, 0, 1, 2, 2, 0, 2, 2, 0, 2, 2, 2, 1, 1, 0, 0, 2, 0, 2, 2, 2, 2, 0, 0, 3, 2, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 0, 4, 1, 2, 2, 0, 2, 1, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 3, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 1, 2, 2, 2, 4, 0, 0, 2, 0, 2, 2, 1, 2, 0, 0

Offset

0,4

Comments

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

Convolution product of A133100 and A133101. - Michael Somos, Feb 10 2015

Links

Table of n, a(n) for n=0..104.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Euler transform of period 10 sequence [ 1, 0, 1, 0, -2, 0, 1, 0, 1, -2, ...].

a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(5^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).

G.f.: Product_{k>0} (1 - x^(5*k))^2 * (1 + x^k) / (1 + x^(5*k)).

G.f.: Sum_{k>=0} a(k) * x^(4k+1) = Sum_{k>0 odd} x^k * (1 - x^(2*k)) * (1 - x^(6*k)) / (1 + x^(10*k)).

Expansion of f(x, x^4) * f(x^2, x^3) in powers of x where f() is the Ramanujan two-variable theta function.

Expansion of psi(x)^2 - x * psi(x^5)^2 in powers of x where psi() is a Ramanujan theta function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133573.

a(n) = A053694(4*n) = A094247(4*n + 1).

a(3*n + 2) = a(5*n + 1) = a(n). - Michael Somos, Feb 10 2015

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/5 = 1.256637... (A019694). - Amiram Eldar, Dec 29 2023

Example

G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + x^11 + ...
G.f. = q + q^5 + q^9 + 2*q^13 + 2*q^17 + q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^5]^3 / (QPochhammer[ x] QPochhammer[ x^10]), {x, 0, n}]; (* Michael Somos, Feb 10 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^5 + A)^3 / (eta(x + A) * eta(x^10 + A)), n))};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 4*n + 1; A=factor(n); prod(k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==5, 1, if( p%4==1, e+1, (1 + (-1)^e) / 2))))))};

Crossrefs

Cf. A019694, A053694, A094247, A133100, A133101, A133573.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A134178 A059018 A249371 * A237291 A146093 A091269

Adjacent sequences: A122187 A122188 A122189 * A122191 A122192 A122193

Keyword

nonn,easy

Author

Michael Somos, Aug 24 2006

Status

approved