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A133574
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Expansion of ( 5 * phi(q^5)^2 - phi(q)^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, 0, -1, 3, 0, 0, -1, -1, 3, 0, 0, -2, 0, 0, -1, -2, -1, 0, 3, 0, 0, 0, 0, 7, -2, 0, 0, -2, 0, 0, -1, 0, -2, 0, -1, -2, 0, 0, 3, -2, 0, 0, 0, 3, 0, 0, 0, -1, 7, 0, -2, -2, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, -1, 6, 0, 0, -2, 0, 0, 0, -1, -2, -2, 0, 0, 0, 0, 0, 3, -1, -2, 0, 0, 6, 0, 0, 0, -2, 3, 0, 0, 0, 0, 0, 0, -2, -1, 0, 7, -2, 0, 0, -2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of psi(-q)^2 * chi(q) * chi(q^5) in powers of q where psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^4) * eta(q^10)^2 / ( eta(q^5) * eta(q^20) ) in powers of q.
Euler transform of period 20 sequence [ -1, -1, -1, -2, 0, -1, -1, -2, -1, -2, -1, -2, -1, -1, 0, -2, -1, -1, -1, -2, ...].
Moebius transform is period 20 sequence [ -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4), A(x^8)) where f(u1, u2, u4, u8) = (u1 - u2)^2 * (u4 - 2*u8)^2 - u2 * u4 * (u2 - u4) * (u2 - 2*u4).
a(n) = -b(n) where b(n) is multiplicative with b(2^e) = 1, b(5^e) = 1-4*e, b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 10 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A053694.
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(4*k)) * (1 + x^(5*k)) / (1 + x^(10*k)).
G.f.: 1 - (Sum_{k>0} x^k / (1 + x^(2*k)) - 5 * x^(5*k) / (1 + x^(10*k))).
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EXAMPLE
| 1 - q - q^2 - q^4 + 3*q^5 - q^8 - q^9 + 3*q^10 - 2*q^13 - q^16 + ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, if( d%5==0, kronecker( -4, d/5) * 5) - kronecker( -4, d)))}
(PARI) {a(n) = local(A, p, e); 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 == 2, 1, if(p == 5, 1 - 4*e, if(p%4 == 1, e + 1, !(e%2) ))))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A)^2 / eta(x^5 + A) / eta(x^20 + A), n))}
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CROSSREFS
| A133573(n) = (-1)^n * a(n).
Sequence in context: A100655 A079275 A133573 * A151859 A163541 A165974
Adjacent sequences: A133571 A133572 A133573 * A133575 A133576 A133577
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 17 2007
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