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A138805
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Theta series of quadratic form x^2 + x*y + 7*y^2.
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1
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1, 2, 0, 0, 2, 0, 0, 4, 0, 6, 0, 0, 0, 4, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 6, 4, 0, 0, 4, 0, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 12, 2, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 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) := 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
| Expansion of theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) in powers of q.
Expansion of phi(q) * phi(q^27) + 4 * q^7 * psi(q^2) * psi(q^54) in powers of q where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 27 sequence [ 2, -2, -2, 2, -2, 2, 2, -2, 6, 2, -2, -2, 2, -2, 2, 2, -2, -6, 2, -2, -2, 2, -2, 2, 2, -2, 0, ...].
a(n) = 2*b(n) where b(n) is multiplicative and b(3^e) = 3 if e>1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(1/2) (t/i) f(t) where q = exp(2 pi i t).
a(3*n + 2) = a(4*n + 2) = 0.
G.f.: Sum_{i,j} x^(i*i + i*j + 7*j*j).
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EXAMPLE
| 1 + 2*q + 2*q^4 + 4*q^7 + 6*q^9 + 4*q^13 + 2*q^16 + 4*q^19 + 2*q^25 + ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, 2 * ( sumdiv(n, d, kronecker(-3, d)) - if( n%3==0, sumdiv(n/3, d, [0, 1, -1, -3, 1, -1, 3, 1, -1][d%9+1]))))}
(PARI) {a(n) = if( n<1, n==0, 2 * if( n%3 == 2, 0, if( n%3==1, sumdiv(n, d, kronecker(-3, d)), if( n%9==0, 3 * sumdiv(n/9, d, kronecker(-3, d))))))}
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 14], n, 1)), n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^54 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^27 + A) * eta(x^108 + A))^2 + 4 * x^7 * (eta(x^4 + A) * eta(x^108 + A))^2 /(eta(x^2 + A) * eta(x^54 + A)), n))}
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CROSSREFS
| 2 * A138806(n) = a(n) unless n=0. A004016(n) = a(9*n).
Sequence in context: A033719 A171608 A024164 * A061897 A047919 A101670
Adjacent sequences: A138802 A138803 A138804 * A138806 A138807 A138808
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Mar 30 2008
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