The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A193514 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A193514

Expansion of phi(-q)^2 * phi(-q^9) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
(history; published version)

#14 by Charles R Greathouse IV at Sun Feb 16 08:33:15 EST 2025

LINKS

Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

Discussion

Sun Feb 16

08:33

OEIS Server: https://oeis.org/edit/global/3014

#13 by Charles R Greathouse IV at Fri Mar 12 22:24:46 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

#12 by N. J. A. Sloane at Wed Nov 13 21:54:14 EST 2019

LINKS

M. Somos, <a href="http://cis.csuohio.edu/~somosA010815/multiqa010815.pdftxt">Introduction to Ramanujan theta functions</a>

Discussion

Wed Nov 13

21:54

OEIS Server: https://oeis.org/edit/global/2830

#11 by Wesley Ivan Hurt at Wed Apr 04 13:16:32 EDT 2018

STATUS

reviewed

approved

#10 by Michel Marcus at Wed Apr 04 00:30:35 EDT 2018

STATUS

proposed

reviewed

#9 by G. C. Greubel at Tue Apr 03 23:57:54 EDT 2018

STATUS

editing

proposed

#8 by G. C. Greubel at Tue Apr 03 23:57:45 EDT 2018

LINKS

G. C. Greubel, <a href="/A193514/b193514.txt">Table of n, a(n) for n = 0..1000</a>

STATUS

approved

editing

#7 by Michael Somos at Fri Apr 24 22:12:45 EDT 2015

STATUS

editing

approved

#6 by Michael Somos at Fri Apr 24 22:12:05 EDT 2015

FORMULA

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 432^(1/2) (t / i) g(t) where q = exp(2 pi Pi i t) and g() is g.f. for A193426.

EXAMPLE

G.f. = 1 - 4*q + 4*q^2 + 2*q^3 - 4*q^4 + 4*q^6 - 8*q^7 + 4*q^8 + 2*q^9 + 2*q^12 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^9] / EllipticTheta[ 4, 0, q^3] , , {q, 0, n}];

PROG

(PARI) {a(n) = if( n<1, n==0, 2 * if( n%3==1, -2, 1) * sumdiv( n, d, -(-1)^d * kronecker( -3, d)))};

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^18 + A)), n))};

STATUS

approved

editing

Discussion

Fri Apr 24

22:12

Michael Somos: Light and space edits.

#5 by Charles R Greathouse IV at Wed Apr 30 01:36:12 EDT 2014

AUTHOR

_Michael Somos, _, Jul 29 2011

Discussion

Wed Apr 30

01:36

OEIS Server: https://oeis.org/edit/global/2181