A226007 Expansion of x * f(-x^7) * f(-x^21) / (f(-x) * f(-x^3)) where f() is a Ramanujan theta function.

1, 1, 2, 4, 6, 9, 16, 21, 32, 48, 66, 92, 134, 177, 243, 334, 441, 585, 788, 1018, 1334, 1746, 2240, 2877, 3715, 4698, 5972, 7582, 9517, 11940, 15005, 18639, 23190, 28812, 35544, 43808, 53996, 66084, 80887, 98868, 120278, 146157, 177497, 214562, 259227

Offset

1,3

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 = 1..2500

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^7) * eta(q^21) / (eta(q) * eta(q^3)) in powers of q.

Euler transform of period 21 sequence [1, 1, 2, 1, 1, 2, 0, 1, 2, 1, 1, 2, 1, 0, 2, 1, 1, 2, 1, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (1 + 7 * u*v) - (u+v) * (u^2 - 3 * u*v + v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (21 t)) = 1/7 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226006.

G.f.: x * Product_{k>0} (1 - x^(7*k)) * (1 - x^(21*k)) / ( (1 - x^k) * (1 - x^(3*k))).

Convolution inverse of A226006.

a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 3^(1/4) * 7^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

Example

G.f. = q + q^2 + 2*q^3 + 4*q^4 + 6*q^5 + 9*q^6 + 16*q^7 + 21*q^8 + 32*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^7] QPochhammer[ q^21] / (QPochhammer[ q] QPochhammer[ q^3]), {q, 0, n}]; (* Michael Somos, Apr 12 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^7 + A) * eta(x^21 + A) / (eta(x + A) * eta(x^3 + A)), n))};

Crossrefs

Cf. A226006.

Sequence in context: A350552 A352359 A372542 * A372632 A257655 A318026

Adjacent sequences: A226004 A226005 A226006 * A226008 A226009 A226010

Keyword

nonn

Author

Michael Somos, May 22 2013

Status

approved