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A138506
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Expansion of f(q)^5 / f(q^5) in powers of q where f() is a Ramanujan theta function.
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4
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1, 5, 5, -10, -15, 5, -10, -30, 25, 35, 5, 60, 30, -60, -30, -10, -55, -80, 35, 100, -15, 60, 60, -110, -50, 5, -60, -100, 90, 150, -10, 160, 105, -120, -80, -30, -105, -180, 100, 120, 25, 210, 60, -210, -180, 35, -110, -230, 110, 215, 5, 160, 180, -260, -100
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of eta(q^2)^15 * eta(q^5) * eta(q^20) / (eta(q)^5 * eta(q^4)^5 * eta(q^10)^3) in powers of q.
Euler transform of period 20 sequence [ 5, -10, 5, -5, 4, -10, 5, -5, 5, -8, 5, -5, 5, -10, 4, -5, 5, -10, 5, -4, ...].
a(n) = 5*b(n) where b() is multiplicative with b(2^e) = ((-2)^(n+1) - 1) / 3, b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 2, 3 (mod 5), b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5).
G.f.: Product_{k>0} (1 - (-x)^k)^5 / (1 - (-x)^(5*k)).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(3*sqrt(5)) = 1.471273... . - Amiram Eldar, Jan 29 2024
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EXAMPLE
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G.f. = 1 + 5*q + 5*q^2 - 10*q^3 - 15*q^4 + 5*q^5 - 10*q^6 - 30*q^7 + 25*q^8 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^5 / QPochhammer[ -q^5], {q, 0, n}]; (* Michael Somos, May 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -5 (-1)^n DivisorSum[ n, # KroneckerSymbol[ 5, #] &]]; (* Michael Somos, May 24 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, -5 * (-1)^n * sumdiv(n, d, d * kronecker(5, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(-x + A)^5 / eta(-x^5 + A), n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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