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A122857
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Expansion of (phi(q)^2 + phi(q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function.
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1
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1, 2, 2, 2, 2, 4, 2, 0, 2, 2, 4, 0, 2, 4, 0, 4, 2, 4, 2, 0, 4, 0, 0, 0, 2, 6, 4, 2, 0, 4, 4, 0, 2, 0, 4, 0, 2, 4, 0, 4, 4, 4, 0, 0, 0, 4, 0, 0, 2, 2, 6, 4, 4, 4, 2, 0, 0, 0, 4, 0, 4, 4, 0, 0, 2, 8, 0, 0, 4, 0, 0, 0, 2, 4, 4, 6, 0, 0, 4, 0, 4, 2, 4, 0, 0, 8, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0, 2, 4, 2, 0, 6, 4, 4, 0, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (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 eta(q^2)^3 * eta(q^3)^2 * eta(q^6) / (eta(q)^2 * eta(q^4 )* eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 2, -1, 0, 0, 2, -4, 2, 0, 0, -1, 2, -2, ...].
Moebius transform is period 12 sequence [ 2, 0, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, ...].
a(12*n + 7) = a(12*n + 11) = 0.
a(n) = 2 * b(n) where b(n) is multiplicative and b(2^e) = b(3^e) = 1, b(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) == (1-(-1)^e)/2 if p == 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A125061.
A035154(n) = a(n) / 2 if n>0. A008441(n) = a(4*n + 1) / 2. A125079(n) = a(2*n + 1) / 2. A113446(3*n + 1) = A002654(3*n + 1) = a(3*n + 1) / 2.
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PROG
| (PARI) {a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -36, d)))}
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p<5, 1, if( p%12<6, e+1, !(e%2))))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)), n))}
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CROSSREFS
| Cf. A002654, A008441, A035154, A113446, A1215079, A125061.
Sequence in context: A092904 A062816 * A132003 A109810 A122066 A053238
Adjacent sequences: A122854 A122855 A122856 * A122858 A122859 A122860
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Sep 14 2006
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