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A133657
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Expansion of q * ( phi(q) * psi(q^4) )^2 in powers of q where phi(), psi() are Ramanujan theta functions.
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1
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1, 4, 4, 0, 6, 16, 8, 0, 13, 24, 12, 0, 14, 32, 24, 0, 18, 52, 20, 0, 32, 48, 24, 0, 31, 56, 40, 0, 30, 96, 32, 0, 48, 72, 48, 0, 38, 80, 56, 0, 42, 128, 44, 0, 78, 96, 48, 0, 57, 124, 72, 0, 54, 160, 72, 0, 80, 120, 60, 0, 62, 128, 104, 0, 84, 192, 68, 0, 96, 192, 72, 0, 74, 152
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 ( eta(q^2)^5 * eta(q^8)^2 / ( eta(q)^2 * eta(q^4)^3 ) )^2 in powers of q.
Euler transform of period 8 sequence [ 4, -6, 4, 0, 4, -6, 4, -4, ...].
a(n) is multiplicative and a(2) = 4, a(2^e) = 0 if e>1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 (t/i)^2 g(t) where q = exp(2 pi i t) and g() is g.f. for A133690.
a(4*n) = 0. a(4*n+2) = 4 * sigma(2*n+1). a(2*n+1) = sigma(2*n+1).
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EXAMPLE
| q + 4*q^2 + 4*q^3 + 6*q^5 + 16*q^6 + 8*q^7 + 13*q^9 + 24*q^10 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, if( n%2, sigma(n), if( n%4, 4 * sigma(n/2), 0)))}
(PARI) {a(n) = local(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^5 * eta(x^8 + A)^2 / eta(x + A)^2 / eta(x^4 + A)^3 )^2, n))}
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CROSSREFS
| Convolution square of A113411. -(-1)^n * A121455(n) = a(n). A008438(n) = a(2*n+1). A112610(n) = a(4*n+1). 4 * A097723(n) = a(4*n+3).
Sequence in context: A016705 A169783 A121455 * A200519 A129507 A021698
Adjacent sequences: A133654 A133655 A133656 * A133658 A133659 A133660
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KEYWORD
| nonn,mult
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AUTHOR
| Michael Somos, Sep 20 2007
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