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A138515
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Expansion of q^(-1/4) * eta(q^2)^8 / (eta(q) * eta(q^4))^2 in powers of q.
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1
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1, 2, -3, -6, 2, 0, -1, 10, 0, 2, 10, -6, -7, -14, 0, 10, -12, 0, -6, 0, 9, 4, 10, 0, 18, 2, 0, -6, -14, 18, -11, -12, 0, 0, -22, 0, 20, -14, -6, -22, 0, 0, 23, 26, 0, 18, 4, 0, -14, 2, 0, 20, 0, 0, 0, -12, 3, -30, 26, 0, -30, -14, 0, 0, 2, -30, -28, 26, 0, 18, 10, 0, -13, 34, 0, 0, 20, 0, 26, -22, 0, 6, 0, -6, 18, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
W. Stein, Modular Forms Database.
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FORMULA
| Coefficients of L-series for elliptic curve "64a4": y^2 = x^3 + x.
Expansion of f(q)^2 * f(-q^2)^2 = psi(-q)^2 * phi(q)^2 = chi(q)^2 * f(-q^2)^4 = psi(q)^2 * phi(-q^2)^2 = f(q)^4 / chi(q)^2 = f(q)^6 / phi(q)^2 = f(-q^2)^6 / psi(-q)^2 = phi(q)^4 / chi(q)^6 = chi(q)^6 * psi(-q)^4 = f(q)^3 * psi(-q) = f(-q^2)^3 * phi(q) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 2, -6, 2, -4, ...].
G.f. is Fourier series of a weight 2 level 64 modular form. f(-1 / (64 t)) = 64 (t/i)^2 f(t) where q = exp(2 pi i t).
a(n) = b(4*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1+(-1)^e)/2 * (-p)^(e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p == 1 (mod 4) with b(p) = 2 * x * (-1)^((x-1)/2) when p = x^2 + 4 * y^2.
a(9*n+2) = -3 * a(n), a(9*n+5) = a(9*n+8) = 0.
G.f.: (Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1)))^2.
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EXAMPLE
| q + 2*q^5 - 3*q^9 - 6*q^13 + 2*q^17 - q^25 + 10*q^29 + 2*q^37 + 10*q^41 + ...
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PROG
| (PARI) {a(n) = ellak( ellinit([0, 0, 0, 1, 0], 1), 4*n + 1)}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A))^2, n))}
(PARI) {a(n) = local(A, p, e, y, a0, a1); 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%4==1, forstep( x=1, sqrtint(p), 2, if( issquare( p - x^2), y=x; break)); y = 2 * y * (2 - (y%4)); a0 = 1; a1 = y; for(i=2, e, x = y * a1 - p * a0; a0=a1; a1=x); a1, if( e%2==0, (-p)^(e / 2)))))))}
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CROSSREFS
| (-1)^n * A002171(n) = a(n). Convolution square of A138514.
Sequence in context: A084459 A093095 A002171 * A107410 A132041 A153634
Adjacent sequences: A138512 A138513 A138514 * A138516 A138517 A138518
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 22 2008
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