The On-Line Encyclopedia of Integer Sequences (OEIS)

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A132977

Expansion of q^(-1/3) * (eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^2 in powers of q.
(history; published version)

#15 by Charles R Greathouse IV at Fri Mar 12 22:24:44 EST 2021

LINKS

M. Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

Discussion

Fri Mar 12

22:24

OEIS Server: https://oeis.org/edit/global/2897

#14 by N. J. A. Sloane at Wed Nov 13 21:58:48 EST 2019

LINKS

M. Somos, <a href="http://somos.crg4.comA010815/multiqa010815.htmltxt">Introduction to Ramanujan theta functions</a>

Discussion

Wed Nov 13

21:58

OEIS Server: https://oeis.org/edit/global/2832

#13 by Joerg Arndt at Wed Aug 09 03:14:02 EDT 2017

STATUS

proposed

approved

#12 by G. C. Greubel at Wed Aug 09 01:04:10 EDT 2017

STATUS

editing

proposed

#11 by G. C. Greubel at Wed Aug 09 01:04:00 EDT 2017

LINKS

G. C. Greubel, <a href="/A132977/b132977.txt">Table of n, a(n) for n = 0..1000</a>

STATUS

approved

editing

#10 by Michael Somos at Sat Oct 31 18:53:25 EDT 2015

STATUS

editing

approved

#9 by Michael Somos at Sat Oct 31 18:52:49 EDT 2015

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

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

LINKS

M. Somos, <a href="http://cis.csuohio.edu/~somos.crg4.com/multiq.pdfhtml">Introduction to Ramanujan theta functions</a>

FORMULA

a(n) = A132975(3*n + 1).

EXAMPLE

q G.f. = 1 + 2*q^4 x + 5*qx^7 2 + 12*qx^10 3 + 26*qx^13 4 + 50*qx^16 5 + 92*qx^6 + 168*x^19 7 + ...

G.f. = q + 2*q^4 + 5*q^7 + 12*q^10 + 26*q^13 + 50*q^16 + 92*q^19 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[(QPochhammer[ x^6]^4 / (QPochhammer[ x] QPochhammer[ x^3] QPochhammer[ x^4] QPochhammer[ x^12]))^2, {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)

PROG

(PARI) {a(n) = localmy(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^6 + A)^4 / (eta(x + A) / * eta(x^3 + A) / * eta(x^4 + A) / * eta(x^12 + A)))^2, n))};

CROSSREFS

A132975(3*n+1) = a(n).

Cf. A112173, A128758, A132975.

STATUS

approved

editing

Discussion

Sat Oct 31

18:53

Michael Somos: Added more info. Light and space edits. Revised Ramanujan theta comment. Updated URL.

#8 by Vaclav Kotesovec at Tue Sep 08 08:14:43 EDT 2015

STATUS

editing

approved

#7 by Vaclav Kotesovec at Tue Sep 08 08:14:12 EDT 2015

FORMULA

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

#6 by Vaclav Kotesovec at Tue Sep 08 08:11:52 EDT 2015

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[((1-x^(6*k))^4 / ( (1-x^k) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(12*k)) ))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)

STATUS

approved

editing