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A138483
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Expansion of (phi(q)^3 * phi(q^5) - phi(q) * phi(q^5)^3) / 4 in powers of q where phi() is a Ramanujan theta function.
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0
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1, 3, 2, 1, 5, 6, 6, 7, 7, 15, 12, 2, 12, 18, 10, 9, 16, 21, 20, 5, 12, 36, 22, 14, 25, 36, 20, 6, 30, 30, 32, 23, 24, 48, 30, 7, 36, 60, 24, 35, 42, 36, 42, 12, 35, 66, 46, 18, 43, 75, 32, 12, 52, 60, 60, 42, 40, 90, 60, 10, 62, 96, 42, 41, 60, 72, 66, 16, 44, 90, 72, 49, 72, 108
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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
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FORMULA
| Expansion of q * phi(q) * chi(q) * f(q^5) * f(-q^10)^2 in powers of q where phi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^7 * eta(q^10)^5 / (eta(q)^3 * eta(q^4)^3 * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 3, -4, 3, -1, 4, -4, 3, -1, 3, -8, 3, -1, 3, -4, 4, -1, 3, -4, 3, -4, ...].
a(n) is multiplicative with a(2^e) = (2^(e+1) - 5*(-1)^e) / 3 if e>0, a(5^e) = 5^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), a(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 2, 3 (mod 5).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 80^(1/2) (t/i)^2 g(t) where q = exp(2 pi i t) and g() is g.f. for A113185.
G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^3 * (1 - x^(5*k))^3 * (1 + x^(10*k-5))^4 * (1 + x^(10*k))^3.
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EXAMPLE
| q + 3*q^2 + 2*q^3 + q^4 + 5*q^5 + 6*q^6 + 6*q^7 + 7*q^8 + 7*q^9 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, n/d * kronecker(25, d) * (-1)^((d+2) \ 5) * if(d%4, 1, 5)))}
(PARI) {a(n) = local(A, p, e, f); 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, (2^(e+1) - 5*(-1)^e) / 3, f = kronecker(5, p); (p^(e+1) - f^(e+1)) / (p - f)))))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^10 + A)^5 / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^5 + A) * eta(x^20 + A)), n))}
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CROSSREFS
| A110712(n) = (-1)^(n+1) * a(n).
Sequence in context: A192022 A139377 A110712 * A065366 A092879 A073370
Adjacent sequences: A138480 A138481 A138482 * A138484 A138485 A138486
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KEYWORD
| nonn,mult
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AUTHOR
| Michael Somos, Mar 20 2008
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