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A138507
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Expansion of (f(q)^5 / f(q^5) - 1) / 5 in powers of q where f() is a Ramanujan theta function.
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1
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1, 1, -2, -3, 1, -2, -6, 5, 7, 1, 12, 6, -12, -6, -2, -11, -16, 7, 20, -3, 12, 12, -22, -10, 1, -12, -20, 18, 30, -2, 32, 21, -24, -16, -6, -21, -36, 20, 24, 5, 42, 12, -42, -36, 7, -22, -46, 22, 43, 1, 32, 36, -52, -20, 12, -30, -40, 30, 60, 6, 62, 32, -42, -43, -12, -24, -66, 48, 44, -6, 72, 35, -72, -36, -2, -60, -72, 24
(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
| a(n) is multiplicative with a(2^e) = ((-2)^(e+1) - 1) / 3, a(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 7 (mod 10), a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10).
G.f.: (Product_{k>0} (1 - (-x)^k)^5 / (1 - (-x)^(5*k)) - 1) / 5.
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EXAMPLE
| q + q^2 - 2*q^3 - 3*q^4 + q^5 - 2*q^6 - 6*q^7 + 5*q^8 + 7*q^9 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, d * kronecker(5, d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(-x + A)^5 / eta(-x^5 + A) - 1) / 5, n))}
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CROSSREFS
| -(-1)^n * A109091(n) = a(n). A138506(n) = 5 * a(n) unless n=0.
Sequence in context: A128255 A154948 A109091 * A205699 A109200 A158909
Adjacent sequences: A138504 A138505 A138506 * A138508 A138509 A138510
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Mar 21 2008
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