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Search: id:A113411
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| A113411 |
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Excess of number of divisors of 2n+1 of form 8k+1,8k+3 over those of form 8k+5,8k+7. |
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+0
5
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| 1, 2, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 1, 4, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 1, 4, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 5, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 3, 4, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 0, 2, 0, 0, 1, 6, 0, 0, 4, 2, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 4
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.55).
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FORMULA
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Expansion of phi(q)psi(q^4) in powers of q where psi(), phi() are Ramanujan theta functions.
Expansion of q^(-1)(eta(q^4)^5*eta(q^16)^2)/(eta(q^2)^2*eta(q^8)^3) in powers of q^2.
a(n) = b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8).
Euler transform of period 8 sequence [2, -3, 2, 0, 2, -3, 2, -2, ...].
G.f.: (Sum_{k} x^k^2)(Sum_{k>=0} x^(2k^2+2k)).
G.f.: Sum_{k>=0} a(k)x^(2k+1) = Sum_{k>=0} F(x^(2k+1), x^(3(2k+1))) where F(x, y) = (x+y)/(1+xy).
a(4n+2)=a(4n+3)=0.
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EXAMPLE
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q +2*q^3 +3*q^9 +2*q^11 +2*q^17 +2*q^19 +q^25 +4*q^27 +...
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PROGRAM
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(PARI) a(n)=if(n<0, 0, n=2*n+1; sumdiv(n, d, (-1)^(d%8>3)))
(PARI) {a(n)=local(n1); if(n<0, 0, n1=sqrtint(n); polcoeff( sum(k=1, n1, 2*x^k^2, 1+x*O(x^n))*sum(k=0, n1, x^(2*k^2+2*k)), n))}
(PARI) {a(n)=local(A); if(n<0, 0, 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, n))}
(PARI) {a(n)=local(A, p, e); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(abs(p%8-6)==1, (1+(-1)^e)/2, e+1)))))}
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CROSSREFS
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Cf. A037761(n) = a(4n+1)/2.
Sequence in context: A131636 A077888 A167634 this_sequence A125095 A143161 A142886
Adjacent sequences: A113408 A113409 A113410 this_sequence A113412 A113413 A113414
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Oct 29 2005
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