This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A129451 Expansion of psi(-q)* phi(q^3)/ chi(q) in powers of q where phi(), psi(), chi() are Ramanujan theta functions. 1

%I

%S 1,-2,2,-2,1,-2,2,-2,3,0,2,-2,2,-2,0,-4,2,-2,2,0,1,-2,4,-2,0,-2,2,-2,

%T 3,-2,2,0,2,-2,0,-2,4,-2,2,0,2,-4,0,-4,0,-2,2,-2,1,0,4,-2,2,0,2,-2,2,

%U -4,2,0,3,-2,2,-2,0,0,2,-4,2,0,2,-4,2,-2,0,0,2,-2,4,-2,4,-2,0,-2,0,-4,0,-2,1,0,2,-2,4,-4,0,-2,2,0,4,0,2,-2,2,-2,1

%N Expansion of psi(-q)* phi(q^3)/ chi(q) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

%C 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) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%H M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">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 q^(-1/6)* eta(q)^2* eta(q^4)^2* eta(q^6)^5/ (eta(q^2)^3* eta(q^3)^2* eta(q^12)^2) in powers of q.

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

%F a(n)= b(6n+1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 5, 11 (mod 12), b(p^e) = e+1 if p == 1 (mod 12), b(p^e) = (-1)^e* (e+1) if p == 7 (mod 12).

%e q - 2*q^7 + 2*q^13 - 2*q^19 + q^25 - 2*q^31 + 2*q^37 - 2*q^43 + 3*q^49 +...

%o (PARI) {a(n)= if(n<0, 0, n= 6*n+1; sumdiv(n,d, kronecker(-4,d)*kronecker(12,n/d)))}

%o (PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)^2* eta(x^4+A)^2* eta(x^6+A)^5/ (eta(x^2+A)^3* eta(x^3+A)^2* eta(x^12+A)^2), n))}

%Y Cf. A129449(3n)= a(n).

%K sign

%O 0,2

%A Michael Somos, Apr 16 2007

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Puzzles | Hot | Classics
Recent Additions | More pages | Superseeker | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .