|
| |
|
|
A132069
|
|
Expansion of eta(q)* eta(q^2)^2* eta(q^5)^3/ eta(q^10)^2 in powers of q.
|
|
1
| |
|
|
1, -1, -3, 2, 1, -1, 6, 6, -7, -7, -3, -12, -2, 12, 18, 2, 9, 16, -21, -20, 1, -12, -36, 22, 14, -1, 36, 20, -6, -30, 6, -32, -23, 24, 48, 6, 7, 36, -60, -24, -7, -42, -36, 42, 12, -7, 66, 46, -18, -43, -3, -32, -12, 52, 60, -12, 42, 40, -90, -60, -2, -62, -96, 42, 41, 12, 72, 66, -16, -44, 18, -72, -49, 72, 108, 2, 20, 72
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
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).
|
|
|
REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 253 Eq. (8.12)
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
FORMULA
| Expansion of (5* phi(-q)* phi(-q^5)^3 -phi(-q)^3* phi(-q^5))/4 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 10 sequence [ -1, -3, -1, -3, -4, -3, -1, -3, -1, -4, ...].
a(n)= -b(n) where b(n) is multiplicative with b(5^e) = 1, b(2^e) = 2-((-2)^(e+1)-1)/(-2-1), b(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 9 (mod 10), b(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 7 (mod 10).
G.f.: Product_{k>0} (1-x^k)* (1-x^(2k))^2* (1-x^(5k))/ (1+x^(5k))^2.
G.f.: 1 + Sum_{k>0} (-1)^k* k* x^k/(1-x^k)* kronecker(5,k).
G.f. is a Fourier series which satisfies f(-1/(10 t)) = 2000^(1/2) (t/i)^2 g(t) where q = exp(2 pi i t) and g() is g.f. for A129303.
|
|
|
EXAMPLE
| 1 - q - 3*q^2 + 2*q^3 + q^4 - q^5 + 6*q^6 + 6*q^7 - 7*q^8 - 7*q^9 +...
|
|
|
PROG
| (PARI) {a(n)= if(n<1, n==0, sumdiv(n, d, kronecker(5, d)* d*(-1)^d))}
(PARI) {a(n)=local(A, p, e, a1); if(n<1, n==0, A=factor(n); -prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==5, 1, if(p>2, p*=kronecker(5, p); (p^(e+1)-1)/(p-1), (5+(-2)^(e+1))/3)))))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)* eta(x^2+A)^2* eta(x^5+A)^3/ eta(x^10+A)^2, n))}
|
|
|
CROSSREFS
| A113185(n)= (-1)^n*a(n).
Sequence in context: A176669 A058280 A113185 * A073201 A118654 A111760
Adjacent sequences: A132066 A132067 A132068 * A132070 A132071 A132072
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Michael Somos, Aug 08 2007, Mar 20 2008
|
| |
|
|
