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

0,3

Comments

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

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

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

Example

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 + ...

Mathematica

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

Prog

(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))}

Crossrefs

Cf. A000700, A022597, A073252, A226635.

Sequence in context: A096818 A282467 A318980 * A332302 A287209 A038099

Adjacent sequences: A226619 A226620 A226621 * A226623 A226624 A226625

Keyword

nonn,easy

Author

Michael Somos, Aug 31 2013

Status

approved