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A113260
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Expansion of (-1+psi(q)^5/psi(q^5)-25q^2 psi(q)psi(q^5)^3)/5 in powers of q where psi(q) is a Ramanujan theta function.
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1
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1, -3, -2, 5, 1, 6, -6, -11, 7, -3, 12, -10, -12, 18, -2, 21, -16, -21, 20, 5, 12, -36, -22, 22, 1, 36, -20, -30, 30, 6, 32, -43, -24, 48, -6, 35, -36, -60, 24, -11, 42, -36, -42, 60, 7, 66, -46, -42, 43, -3, 32, -60, -52, 60, 12, 66, -40, -90, 60, -10, 62, -96, -42, 85, -12, 72, -66, -80, 44, 18, 72, -77, -72, 108
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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).
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 249 Entry 8(iv).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| a(n) is multiplicative and a(2^e) = ((-2)^(e+2)-1)/3, a(5^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 4 (mod 5), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 2, 3 (mod 5).
G.f.: Sum_{k>0} k x^k/(1+x^k) kronecker(5, k).
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PROG
| (PARI) a(n)=if(n<1, 0, -sumdiv(n, d, d*kronecker(5, d)*(-1)^(n/d)))
(PARI) {a(n)=local(A, p, e); if(n<1, 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, ((-2)^(e+2)-1)/3, p*=kronecker(5, p); (p^(e+1)-1)/(p-1))))))}
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CROSSREFS
| Cf. A113259(n)=5 a(n) if n>0.
Sequence in context: A121490 A197293 A099643 * A051543 A129538 A076934
Adjacent sequences: A113257 A113258 A113259 * A113261 A113262 A113263
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Oct 20 2005
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