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A121613
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Sum of divisors of 2n+1 multiplied by (-1)^n.
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0
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1, -4, 6, -8, 13, -12, 14, -24, 18, -20, 32, -24, 31, -40, 30, -32, 48, -48, 38, -56, 42, -44, 78, -48, 57, -72, 54, -72, 80, -60, 62, -104, 84, -68, 96, -72, 74, -124, 96, -80, 121, -84, 108, -120, 90, -112, 128, -120, 98, -156, 102, -104, 192, -108, 110, -152, 114, -144, 182, -144, 133, -168, 156, -128
(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
| Euler transform of period 4 sequence [ -4, 0, -4, -4, ...].
Expansion of q^(-1/2)(eta(q)eta(q^4)/eta(q^2))^4 = psi(-q)^4 in powers of q where psi is a Ramanujan theta function.
a(n)=b(2n+1) where b(n) is multiplicative and b(2^e)=0^n, b(p^e)=(p^(e+1)-1)/(p-1) if p == 1 (mod 4), b(p^e)=(-1)^e*(p^(e+1)-1)/(p-1) if p == 3 (mod 4).
G.f.: (Product_{k>0} (1-x^k)/(1-x^(4k-2)))^4 = Sum_{k>0} -(-1)^k*(2k-1)*x^(k-1)/(1+x^(2k-1)).
G.f.: Sum_{k>=0} a(k)x^(2k+1) = x(Prod_{k>0} (1-x^(4k-2))*(1-x^(8k)))^4 = x(Sum_{k>0} (-1)^[k/2] x^(k^2-k))^4 = Sum_{k>=0} (-1)^k*(2k+1)*x^(2k+1)/(1+x^(4k+2)).
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PROG
| (PARI) {a(n)=if(n<0, 0, (-1)^n*sigma(2*n+1))}
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CROSSREFS
| a(n)=(-1)^n*A008438(n).
Sequence in context: A020153 A151760 A008438 * A141641 A145284 A023560
Adjacent sequences: A121610 A121611 A121612 * A121614 A121615 A121616
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Aug 10 2006
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