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A138811
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Theta series of quadratic form x^2 + x*y + 11*y^2.
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0
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1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 4, 0, 4, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 2, 4, 0, 0, 4, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 4, 2, 4, 0, 4, 0
(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 phi(q) * phi(q^43) + 4 * q^11 * psi(q^2) * psi(q^86) in powers of q where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 43 sequence [ 2, -2, -2, 2, -2, 2, -2, -2, 2, 2, 2, -2, 2, 2, 2, 2, 2, -2, -2, -2, 2, -2, 2, 2, 2, -2, -2, -2, -2, -2, 2, -2, -2, -2, 2, 2, -2, 2, -2, 2, 2, -2, 0, ...].
a(n) = 2*b(n) where b(n) is multiplicative and b(43^e) = 1, b(p^e) = e+1 if kronecker(-43, p) = 1, b(p^e) = (1 + (-1)^e) / 2 if kronecker(-43, p) = -1.
G.f. is a period 1 Fourier series which satisfies f(-1 / (43 t)) = 43^(1/2) (t/i) f(t) where q = exp(2 pi i t).
a(4*n + 2) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n) = a(9*n) = a(n).
G.f.: Sum_{i,j} x^(i*i + i*j + 11*j*j).
Expansion of theta_3(q) * theta_3(q^43) + theta_2(q) * theta_2(q^43) in powers of q.
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EXAMPLE
| 1 + 2*q + 2*q^4 + 2*q^9 + 4*q^11 + 4*q^13 + 2*q^16 + 4*q^17 + 4*q^23 + ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-43, d))*2)}
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 22], n, 1)), n))}
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CROSSREFS
| 2 * A035147(n) = a(n) unless n=0.
Sequence in context: A033729 A033725 A033723 * A107494 A079205 A107497
Adjacent sequences: A138808 A138809 A138810 * A138812 A138813 A138814
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Mar 31 2008, Apr 05 2008
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