|
| |
|
|
A080015
|
|
Expansion of theta_3(q)/theta_3(q^2) in powers of q.
|
|
1
| |
|
|
1, 2, -2, -4, 6, 8, -12, -16, 22, 30, -40, -52, 68, 88, -112, -144, 182, 228, -286, -356, 440, 544, -668, -816, 996, 1210, -1464, -1768, 2128, 2552, -3056, -3648, 4342, 5160, -6116, -7232, 8538, 10056, -11820, -13872, 16248, 18996, -22176, -25844
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
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) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
Euler transform of period 8 sequence [2,-5,2,2,2,-5,2,0,...].
|
|
|
REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 214 Entry 24(ii).
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
FORMULA
| Expansion of (eta(q^2)/eta(q^4))^7(eta(q^8)/eta(q))^2 in powers of q.
G.f.: A(x)/B(x), where A(x) = Sum_{m = -infinity..infinity} x^(m^2) and B(x) = Sum_{m = -infinity..infinity} x^(2*m^2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 22 2005
Expansion of phi(x)/phi(x^2) where phi() is a Ramanujan theta function.
G.f. A(x) satisfies 0=f(A(x),A(x^2)) where f(u,v)=1+(1-u*v)^2-v^2 . - Michael Somos Jan 31 2006
G.f. A(x) satisfies 0=f(A(x),A(x^3)) where f(u,v)=u^4-v^4+8*u*v-6*u*v*(u^2+v^2)+4*(u*v)^3 . - Michael Somos Jan 31 2006
Expansion of sqrt(m) in powers of q where m is the multiplier for the second degree modular equation.
G.f.: Prod_{n>0} ((1-x^(8n-2))(1-x^(8n-6)))^5/((1-x^(8n-1))(1-x^(8n-3))(1-x^(8n-4))(1-x^(8n-5))(1-x^(8n-7)))^2.
|
|
|
PROG
| (PARI) a(n)=local(A, m); if(n<0, 0, m=1; A=1+2*x+O(x^2); while(m<n, m*=2; A=subst(A, x, x^2); A=(1+2*sqrt((A^2-1)/4))/A); polcoeff(A, n))
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^8+A)/eta(x+A))^2* (eta(x^2+A)/eta(x^4+A))^7, n))}
|
|
|
CROSSREFS
| Apart from signs, same as A080054, which is the main entry for this sequence.
Sequence in context: A179999 A147982 A051466 * A080054 A108494 A078578
Adjacent sequences: A080012 A080013 A080014 * A080016 A080017 A080018
|
|
|
KEYWORD
| sign,easy
|
|
|
AUTHOR
| Michael Somos, Jan 20 2003
|
| |
|
|
