A145786 Expansion of q * chi(q^3) * chi(q^5) / (chi(q) * chi(q^15))^2 in powers of q where chi() is a Ramanujan theta function.

1, -2, 3, -5, 7, -10, 14, -20, 27, -36, 48, -63, 82, -106, 137, -175, 222, -280, 352, -439, 546, -676, 834, -1024, 1253, -1528, 1857, -2250, 2718, -3276, 3936, -4718, 5640, -6728, 8006, -9507, 11266, -13324, 15726, -18526, 21786, -25574, 29970, -35064, 40961

Offset

1,2

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=1..45.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q) * eta(q^4) * eta(q^6) * eta(q^10) * eta(q^15) * eta(q^60))^2 / (eta(q^2)^4 * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30)^4) in powers of q.

Euler transform of a period 60 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 Pi i t).

a(n) = - A145728(n) unless n=0. a(n) = -(-1)^n * A123630(n).

Convolution inverse of A145788.

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(2*n/15)) / (2^(7/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

G.f. = q - 2*q^2 + 3*q^3 - 5*q^4 + 7*q^5 - 10*q^6 + 14*q^7 - 20*q^8 + 27*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^3, x^6] QPochhammer[ -x^5, x^10] / (QPochhammer[ -x, x^2] QPochhammer[ -x^15, x^30])^2 , {x, 0, n}]; (* Michael Somos, Sep 03 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) * eta(x^60 + A))^2 / (eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A) * eta(x^30 + A)^4), n))};

Crossrefs

Cf. A123630, A145728, A145788.

Sequence in context: A023893 A065094 A145728 * A094023 A123630 A326977

Adjacent sequences: A145783 A145784 A145785 * A145787 A145788 A145789

Keyword

sign

Author

Michael Somos, Oct 23 2008

Status

approved