|
Search: id:A113185
|
|
|
| A113185 |
|
Expansion of (5phi(q)phi^3(q^5)-phi^3(q)phi(q^5))/4 in powers of q where phi(q) is a Ramanujan theta function. |
|
+0
3
|
|
| 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
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
REFERENCES
|
B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 249 Entry 8(ii).
|
|
FORMULA
|
Expansion of (eta(q^2)^5 eta(q^10)^7)/(eta(q)eta(q^4)eta(q^5)^3 eta(q^20)^3) in powers of q.
Euler transform of period 20 sequence [1, -4, 1, -3, 4, -4, 1, -3, 1, -8, 1, -3, 1, -4, 4, -3, 1, -4, 1, -4, ...].
a(n) is multiplicative with a(5^e) = 1, a(2^e) = ((-2)^(e+1)-1)/(-2-1)-2 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 9 (mod 10), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 7 (mod 10).
G.f.: (5phi(q) phi(q^5)^3 - phi(q)^3 phi(q^5))/4 where phi(q)=1+2(q+q^4+q^9+...).
|
|
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 + ...
|
|
PROGRAM
|
(PARI) a(n)=if(n<1, n==0, (-1)^n*sumdiv(n, d, kronecker(5, d)*d*(-1)^d))
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^10+A)^7/(eta(x+A)*eta(x^4+A)*eta(x^5+A)^3*eta(x^20+A)^3), n))}
(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), a1=-3; for(i=2, e, a1=-2*a1-5); a1)))))}
|
|
CROSSREFS
|
(-1)^n * A132069(n) = a(n).
Sequence in context: A046225 A123396 A058280 this_sequence A132069 A073201 A118654
Adjacent sequences: A113182 A113183 A113184 this_sequence A113186 A113187 A113188
|
|
KEYWORD
|
sign,mult
|
|
AUTHOR
|
Michael Somos, Oct 17 2005, Mar 20 2008
|
|
|
Search completed in 0.002 seconds
|
