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A113447
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Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.
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2
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1, 1, 1, -1, 0, 1, 2, 1, 1, 0, 0, -1, 2, 2, 0, -1, 0, 1, 2, 0, 2, 0, 0, 1, 1, 2, 1, -2, 0, 0, 2, 1, 0, 0, 0, -1, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, -1, 3, 1, 0, -2, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, -1, 0, 0, 2, 0, 0, 0, 0, 1, 2, 2, 1, -2, 0, 2, 2, 0, 1, 0, 0, -2, 0, 2, 0, 0, 0, 0, 4, 0, 2, 0, 0, 1, 2, 3, 0, -1, 0, 0, 2, 2, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,7
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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 (eta(q^2)*eta(q^3)^3*eta(q^12)^3)/(eta(q)*eta(q^4)*eta(q^6)^3) in powers of q.
Euler transform of period 12 sequence [1, 0, -2, 1, 1, 0, 1, 1, -2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0, ...].
a(n) is multiplicative and a(2^e) = -(-1)^e if e>0, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} x^(6k-5)/(1-x^(6k-5)) -x^(6k-1)/(1-x^(6k-1)) -2*x^(12k-8)/(1-x^(12k-8)) +2x^(12k-4)/(1-x^(12k-4)).
G.f.: Sum_{k>0} x^k(1-x^(3k))^2/(1+x^(4k)+x^(8k)).
G.f.: x*Product_{k>0} (1-x^k)/(1-x^(4k-2))*((1-x^(12k-6))/(1-x^(3k)))^3.
Expansion of theta_2(iq^3)^3/(4*theta_2(iq)) in powers of q^2.
Expansion of q * psi(-q^3)^3 / psi(-q) in powers of q where psi() is a Ramanujan theta function.
Expansion of (c(q) * c(q^4)) / (3 * c(q^2)) in powers of q where c() is a cubic AGM function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3)^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A132973.
a(3n) = a(n). a(6n+5) = a(12n+10) = 0.
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EXAMPLE
| q + q^2 + q^3 - q^4 + q^6 + 2*q^7 + q^8 + q^9 - q^12 + 2*q^13 + ...
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PROG
| (PARI) {a(n)=local(x); if(n<1, 0, x=valuation(n, 2); -(-1)^max(1, x)*sumdiv(n, d, kronecker(-12, d)))}
(PARI) {a(n)=if(n<1, 0, direuler(p=2, n, if(p==2, 1+X/(1+X), 1/(1-X)/(1-kronecker(-12, p)*X)))[n])}
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)^3*eta(x^12+A)^3/(eta(x+A)*eta(x^4+A)*eta(x^6+A)^3), n))}
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CROSSREFS
| Convolution inverse of A133637.
A035178(n) = |a(n)|. A093829(n) = a(2n). A033762(n) = a(2n+1). A112604(n) = a(4n+1). A112605(n) = a(4n+3). A097195(n) = a(6n+1). A033687(n) = a(6n+2).
A112606(n) = a(8n+1). A112608(n) = a(8n+3). 2 * A112607(n) = a(8n+5). A112605(n) = a(8n+6). 2 * A112609(n) = a(8n+7).
A123884(n) = a(12n+1). 2 * A121361(n) = a(12n+7). A131961(n) = a(24n+1). 2 * A131962(n) = a(24n+7). 2 * A131963(n) = a(24n+13). 2 * A131964(n) = a(24n+19).
Sequence in context: A035178 A093829 * A137608 A191336 A078807 A029422
Adjacent sequences: A113444 A113445 A113446 * A113448 A113449 A113450
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Nov 02 2005
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