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A138512
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Expansion of eta(q) * eta(q^4) * eta(q^10)^15 / (eta(q^2)^3 * eta(q^5)^5 * eta(q^20)^5) in powers of q.
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0
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1, -1, 2, -3, 5, -2, 6, -5, 7, -5, 12, -6, 12, -6, 10, -11, 16, -7, 20, -15, 12, -12, 22, -10, 25, -12, 20, -18, 30, -10, 32, -21, 24, -16, 30, -21, 36, -20, 24, -25, 42, -12, 42, -36, 35, -22, 46, -22, 43, -25, 32, -36, 52, -20, 60, -30, 40, -30, 60, -30, 62, -32, 42, -43, 60, -24, 66, -48, 44, -30, 72, -35, 72
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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 * f(q^5)^5 / f(q) in powers of q where f() is a Ramanujan theta function.
Euler transform of period 20 sequence [ -1, 2, -1, 1, 4, 2, -1, 1, -1, -8, -1, 1, -1, 2, 4, 1, -1, 2, -1, -4, ...].
a(n) is multiplicative with a(2^e) = -(2^(e+1) - (-1)^(e+1)) / 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).
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EXAMPLE
| q - q^2 + 2*q^3 - 3*q^4 + 5*q^5 - 2*q^6 + 6*q^7 - 5*q^8 + 7*q^9 - 5*q^10 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, d * kronecker(5, n/d)))}
(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) - (-1)^(e+1)) / 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 + A) * eta(x^4 + A) * eta(x^10 + A)^15 / (eta(x^2 + A)^3 * eta(x^5 + A)^5 * eta(x^20 + A)^5), n))}
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CROSSREFS
| (-1)^n * A053723(n) = a(n+1).
Sequence in context: A093870 A126833 A053723 * A201652 A066949 A073481
Adjacent sequences: A138509 A138510 A138511 * A138513 A138514 A138515
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Mar 21 2008
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