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Search: id:A134013
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| A134013 |
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Expansion of q * phi(q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions. |
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3
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| 1, 2, 0, 0, 2, 0, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 6, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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1,2
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FORMULA
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Eqpansion of eta(q^2)^5 * eta(q^16)^2 / ( eta(q)^2 * eta(q^4)^2 * eta(q^8) ) in powers of q.
Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, ...].
Moebius transform is period 16 sequence [ 1, 1, -1, -2, 1, -1, -1, 0, 1, 1, -1, 2, 1, -1, -1, 0, ...].
Multiplicative with a(2) = 2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A134014.
a(4*n) = a(4*n+3) = a(8*n+6) = 0. a(8*n+2) = 2 * a(4*n+1).
G.f.: Sum_{k>0} kronecker(-4, k) * x^k * (1 + x^k)^2 / (1 - x^(4*k)).
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EXAMPLE
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q + 2*q^2 + 2*q^5 + q^9 + 4*q^10 + 2*q^13 + 2*q^17 + 2*q^18 + 3*q^25 + ...
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PROGRAM
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(PARI) {a(n) = if( n>0 & (n+1)%4\2, (n%4) * sumdiv( n/gcd(n, 2), d, (-1)^(d\2)))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / eta(x + A)^2 / eta(x^4 + A)^2 / eta(x^8 + A), n))}
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CROSSREFS
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Cf. A112301(n) = -(-1)^n * a(n). A008441(n) = a(4*n+1). A113407(n) = a(8*n+1). 2 * A053692(n) = a(8*n+5).
Adjacent sequences: A134010 A134011 A134012 this_sequence A134014 A134015 A134016
Sequence in context: A000095 A034949 A112301 this_sequence A136521 A066448 A108497
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KEYWORD
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nonn,mult
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AUTHOR
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Michael Somos, Oct 02 2007
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