A226622 Expansion of phi(x^2) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

1, 1, 4, 5, 9, 13, 21, 29, 46, 62, 90, 122, 171, 227, 311, 408, 545, 709, 933, 1198, 1555, 1980, 2536, 3205, 4063, 5092, 6400, 7966, 9928, 12281, 15198, 18684, 22979, 28097, 34346, 41789, 50813, 61527, 74453, 89757, 108114, 129809, 155704, 186221, 222503

Offset: 0

Author

Michael Somos, Aug 31 2013

Keywords

• nonn
• easy

Comments

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

Examples

			1 + x + 4*x^2 + 5*x^3 + 9*x^4 + 13*x^5 + 21*x^6 + 29*x^7 + 46*x^8 + 62*x^9 + ...
1/q + q^23 + 4*q^47 + 5*q^71 + 9*q^95 + 13*q^119 + 21*q^143 + 29*q^167 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A000700, A022597, A073252, A226635.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] / QPochhammer[ q], {q, 0, n}]

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^2), n))}

Formula

Expansion of q^(1/24) * eta(q^4)^5 / (eta(q) * eta(q^2)^2 * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [ 1, 3, 1, -2, 1, 3, 1, 0, ...].

G.f.: (Sum_{k in Z} x^(2*k^2)) / (Product_{k>0} (1 - x^k)).

a(n) = A022597(2*n) = A073252(2*n).

G.f. A(x) satisfies A(x^2) = ( chi(x)^2 + chi(-x)^2 )/2, where chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700. Cf. A226635. - Peter Bala, Sep 29 2023

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2025