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A226007 Expansion of x * f(-x^7) * f(-x^21) / (f(-x) * f(-x^3)) where f() is a Ramanujan theta function. 2
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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: A074677 A179997 A101756 * A257655 A318026 A173241

Adjacent sequences:  A226004 A226005 A226006 * A226008 A226009 A226010

KEYWORD

nonn

AUTHOR

Michael Somos, May 22 2013

STATUS

approved

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Last modified June 20 17:27 EDT 2019. Contains 324234 sequences. (Running on oeis4.)