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A131947
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Expansion of (1-(phi(-q)* phi(-q^3))^2)/4 in powers of q where phi() is a Ramanujan theta function.
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1
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1, -1, 1, -5, 6, -1, 8, -13, 1, -6, 12, -5, 14, -8, 6, -29, 18, -1, 20, -30, 8, -12, 24, -13, 31, -14, 1, -40, 30, -6, 32, -61, 12, -18, 48, -5, 38, -20, 14, -78, 42, -8, 44, -60, 6, -24, 48, -29, 57, -31, 18, -70, 54, -1, 72, -104, 20, -30, 60, -30, 62, -32, 8, -125, 84, -12, 68, -90, 24, -48, 72, -13, 74, -38, 31
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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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
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.66).
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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 with a(2^e) = 3- 2^(e+1), a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p>3.
G.f.: Sum_{k>0} k (-x)^k/(1-x^k) kronecker(9, k) = ((theta_3(-x)theta_3(-x^3))^2-1)/4.
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PROG
| (PARI) {a(n)= if(n<1, 0, sumdiv(n, d, d*((abs(d%6-3)==2)-(abs(d%6-3)==1))))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( (1-(eta(x+A)* eta(x^3+A))^4/( eta(x^2+A)* eta(x^6+A))^2)/4, n))}
(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==2, 3-p^(e+1), if(p==3, 1, (p^(e+1)-1)/(p-1))))))}
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CROSSREFS
| -(-1)^n*A113262(n) = a(n). A131946(n) = -4*a(n) unless n=0.
Sequence in context: A021182 A175647 A113262 * A195823 A105577 A054655
Adjacent sequences: A131944 A131945 A131946 * A131948 A131949 A131950
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Jul 30 2007
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