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!)
A298933 Expansion of f(x, x^2) * f(x, x^3) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function. 2

%I #11 Mar 12 2021 22:24:48

%S 1,2,3,4,4,6,5,6,6,4,8,6,9,6,6,12,8,12,8,8,9,8,12,6,8,14,12,12,8,12,

%T 13,12,18,8,8,12,16,14,12,12,16,12,13,14,6,20,16,18,8,10,18,16,20,12,

%U 16,16,15,20,12,18,24,14,18,8,16,18,16,22,12,12,20,24

%N Expansion of f(x, x^2) * f(x, x^3) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

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

%H Robert Israel, <a href="/A298933/b298933.txt">Table of n, a(n) for n = 0..10000</a>

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

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

%F Expansion of phi(x) * phi(-x^3) * phi(-x^6) / chi(-x^2)^3 in powers of x where phi(), chi() are Ramanujan theta functions.

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

%F Euler transform of period 12 sequence [2, 0, 0, -1, 2, -3, 2, -1, 0, 0, 2, -3, ...].

%F a(n) = A298932(2*n).

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

%e G.f. = q + 2*q^5 + 3*q^9 + 4*q^13 + 4*q^17 + 6*q^21 + 5*q^25 + 6*q^29 + ...

%p N:= 100:

%p S:= series(JacobiTheta3(0,x)*JacobiTheta4(0,x^3)*JacobiTheta4(0,x^6)*expand(QDifferenceEquations:-QPochhammer(-x^2,x^2,floor(N/2)))^3, x, N+1):

%p seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Jan 29 2018

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2]^3, {x, 0, n}];

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

%Y Cf. A298932.

%K nonn

%O 0,2

%A _Michael Somos_, Jan 29 2018

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 March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)