Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)

Revision History for A263433

(Underlined text is an addition; strikethrough text is a deletion.)

Showing entries 1-10 | older changes
A263433 Expansion of f(x, x) * f(x^2, x^4)^2 in powers of x where f(, ) is Ramanujan's general theta function.
(history; published version)
#11 by Charles R Greathouse IV at Fri Mar 12 22:24:48 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
#10 by N. J. A. Sloane at Wed Nov 13 21:58:51 EST 2019
LINKS

M. Somos, <a href="http://somos.crg4.com="/A010815/multiqa010815.htmltxt">Introduction to Ramanujan theta functions</a>

Discussion
Wed Nov 13 21:58 
OEIS Server: https://oeis.org/edit/global/2832
#9 by Alois P. Heinz at Tue Jul 31 22:33:12 EDT 2018
STATUS

reviewed

approved

#8 by Michel Marcus at Tue Jul 31 13:43:10 EDT 2018
STATUS

proposed

reviewed

#7 by G. C. Greubel at Tue Jul 31 13:42:27 EDT 2018
STATUS

editing

proposed

#6 by G. C. Greubel at Tue Jul 31 13:42:23 EDT 2018
LINKS

G. C. Greubel, <a href="/A263433/b263433.txt">Table of n, a(n) for n = 0..2500</a>

STATUS

approved

editing

#5 by Michael Somos at Sun Oct 18 14:57:26 EDT 2015
STATUS

editing

approved

#4 by Michael Somos at Sun Oct 18 14:57:16 EDT 2015
FORMULA

G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 15552^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263444.

CROSSREFS

Cf. A005889, A045832, A261426, A263444.

STATUS

approved

editing

Discussion
Sun Oct 18 14:57 
Michael Somos: Added more info.
#3 by Michael Somos at Sun Oct 18 10:27:19 EDT 2015
STATUS

editing

approved

#2 by Michael Somos at Sun Oct 18 10:26:31 EDT 2015
NAME

allocatedExpansion of f(x, x) * f(x^2, x^4)^2 in powers of x where f(, ) is Ramanujan's forgeneral Michaeltheta Somosfunction.

DATA

1, 2, 2, 4, 5, 6, 6, 4, 7, 4, 6, 8, 4, 10, 8, 12, 8, 6, 14, 8, 11, 6, 8, 8, 8, 14, 6, 12, 15, 14, 14, 8, 12, 14, 12, 16, 8, 10, 14, 16, 16, 12, 12, 12, 16, 10, 10, 8, 19, 20, 20, 8, 12, 24, 14, 24, 12, 16, 14, 16, 21, 10, 14, 28, 16, 12, 14, 12, 16, 16, 30, 12

OFFSET

0,2

COMMENTS

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

LINKS

M. Somos, <a href="http://somos.crg4.com/multiq.html">Introduction to Ramanujan theta functions</a>

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

FORMULA

Expansion of f(-x^2)^2 * phi(-x^6)^2 / phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.

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

a(n) = A261426(2*n) = A045832(6*n). 3 * a(n) = A005889(6*n).

EXAMPLE

G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 6*x^6 + 4*x^7 + 7*x^8 + ...

G.f. = q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 6*q^37 + 4*q^43 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 4, 0, x], {x, 0, n}];

PROG

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

CROSSREFS

Cf. A005889, A045832, A261426.

KEYWORD

allocated

nonn

AUTHOR

Michael Somos, Oct 18 2015

STATUS

approved

editing

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 18 19:26 EDT 2024. Contains 375273 sequences. (Running on oeis4.)