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

Author

Michael Somos, Aug 24 2006

Keywords

• nonn
• easy

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

Examples

			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 + ...
		

Links

• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

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

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

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

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))))))};

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