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A159818
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Expansion of f(q) * f(q^5) in powers of q where f() is a Ramanujan theta function.
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1
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1, 1, -1, 0, 0, 0, 1, -2, 0, 0, -2, -1, -1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, -2, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, -2, 0, 0, -2, -1, 0, 0, 0, 2, 0, 2, 0, 0, -2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 1, -2, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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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 q^(-1/4) * eta(q^2)^3 * eta(q^10)^3 / (eta(q) * eta(q^4) * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 1, -2, 1, -1, 2, -2, 1, -1, 1, -4, 1, -1, 1, -2, 2, -1, 1, -2, 1, -2, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(5^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3, 7 (mod 20), b(p^e) = (-1)^(e*z) * (e+1) if p == 1, 9 (mod 20) where p = x^2 + 5*y^2 and z = 1 if x or y == 0 (mod 4) else z = 0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (320 t)) = (320)^(1/2) (t/i) f(t) where q = exp(2 pi i t).
G.f.: Product_{k>0} (1 - (-x)^k) * (1 - (-x)^(5*k)).
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EXAMPLE
| q + q^5 - q^9 + q^25 - 2*q^29 - 2*q^41 - q^45 - q^49 + 2*q^61 + q^81 + ...
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PROG
| (PARI) {a(n) = local(A, p, e, x, z); if(n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if(p==2, 0, if(p==5, 1, if(p%20>10, !(e%2), if(p%4==3, kronecker(-4, e+1), for(y=1, sqrtint(p\5), if(issquare(p - 5*y^2, &x), z = if(x%2, y, x)%4/2; break)); (-1)^(e*z) *(e+1))))))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^10 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)), n))}
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CROSSREFS
| A030202(n) = (-1)^n * a(n). Convolutions square is A159817
Sequence in context: A156996 A029304 A030202 * A081827 A100286 A029303
Adjacent sequences: A159815 A159816 A159817 * A159819 A159820 A159821
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Apr 22 2009
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