A128762 Expansion of chi(q) * chi(q^2) / (chi(q^5) * chi(q^10)) in powers of q where chi() is a Ramanujan theta function.

1, 1, 1, 2, 1, 1, 2, 2, 2, 4, 4, 4, 5, 5, 6, 6, 8, 9, 10, 12, 14, 15, 17, 20, 21, 23, 26, 30, 32, 37, 42, 44, 50, 56, 60, 66, 74, 80, 88, 98, 109, 119, 130, 144, 154, 167, 184, 200, 218, 241, 262, 284, 308, 334, 362, 390, 426, 462, 498, 542, 589, 633, 685, 742, 796, 858

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..65.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 40 sequence [ 1, 0, 1, -1, 0, 0, 1, 0, 1, 0, 1, -1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, -1, 1, 0, 1, 0, 1, 0, 0, -1, 1, 0, 1, 0, ...].

Given g.f. A(x), then B(q) = q*A(q^2) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (u^2 + v^2).

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(20*k)) / ( (1 + x^(4*k)) * (1+x^(5*k))).

Convolution inverse of A128763.

a(n) ~ exp(Pi*sqrt(n/5)) / (2^(3/2) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Example

G.f. = 1 + x + x^2 + 2*x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 4*x^9 + ...
G.f. = q + q^3 + q^5 + 2*q^7 + q^9 + q^11 + 2*q^13 + 2*q^15 + 2*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5, - x^5] QPochhammer[ x^10, -x^10]) / (QPochhammer[ x, -x] QPochhammer[ x^2, -x^2]), {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(20*k)) / ( (1 + x^(4*k)) * (1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^8  + A) * eta(x^10 + A) * eta(x^20 + A) / (eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^40 + A)), n))};

Crossrefs

Cf. A128763.

Sequence in context: A029255 A029272 A153904 * A096396 A126307 A320838

Adjacent sequences: A128759 A128760 A128761 * A128763 A128764 A128765

Keyword

nonn

Author

Michael Somos, Mar 25 2007

Status

approved