A110712 Expansion of eta(q)^3 * eta(q^5) * eta(q^10)^2 / eta(q^2)^2 in powers of q.
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
Offset: 1
Examples
G.f. = q - 3*q^2 + 2*q^3 - q^4 + 5*q^5 - 6*q^6 + 6*q^7 - 7*q^8 + 7*q^9 - 15*q^10 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Shaun Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 318, Th. 4.1.
- Michael Somos, Introduction to Ramanujan theta functions, 2019.
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Programs
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Mathematica
a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# # KroneckerSymbol[ 5, n/#] &]]; (* Michael Somos, Jul 12 2012 *) a[ n_] := SeriesCoefficient[ (1/4) (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^5]^3 - EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 4, 0, q^5]), {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
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PARI
{a(n) = if( n<1, 0, sumdiv( n, d, -(-1)^d * d * kronecker( n/d, 5)))}; -
PARI
{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^5 + A) * eta(x^10 + A)^2 / eta(x^2 + A)^2, n))};
Formula
Expansion of (phi(-q) * phi(-q^5)^3 - phi(-q)^3 * phi(-q^5)) / 4 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jul 12 2012
Euler transform of period 10 sequence [-3, -1, -3, -1, -4, -1, -3, -1, -3, -4, ...].
Multiplicative with a(p^e) = p^e if p=5, a(p^e) = -(p^(e+1) - 5*(-1)^e) / (p + 1) if p=2, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 or 9 (mod 10), a(p^e) = (p^(e+1) - (-1)^e) / (p + 1) if p == 3 or 7 (mod 10).
G.f.: Sum_{k>0} Kronecker(k, 5) * x^k / (1 + x^k)^2 = x * Product_{k>0} (1 - x^k)^3 * (1 - x^(5*k)) * (1 - x^(10*k))^2 / (1 - x^(2*k))^2.
a(n) = (-1)^(n+1) * A138483(n). - Amiram Eldar, Jan 28 2024
Comments