Search: id:A134013
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%I A134013
%S A134013 1,2,0,0,2,0,0,0,1,4,0,0,2,0,0,0,2,2,0,0,0,0,0,0,3,4,0,0,2,0,0,0,0,4,0,
%T A134013 0,2,0,0,0,2,0,0,0,2,0,0,0,1,6,0,0,2,0,0,0,0,4,0,0,2,0,0,0,4,0,0,0,0,0,
%U A134013 0,0,2,4,0,0,0,0,0,0,1,4,0,0,4,0,0,0,2,4,0,0,0,0,0,0,2,2,0,0,2,0,0,0,0
%N A134013 Expansion of q * phi(q) * psi(q^8) in powers of q where phi(), psi() 
               are Ramanujan theta functions.
%F A134013 Expansion of eta(q^2)^5 * eta(q^16)^2 / ( eta(q)^2 * eta(q^4)^2 * eta(q^8) 
               ) in powers of q.
%F A134013 Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, 0, 2, 
               -3, 2, -1, 2, -3, 2, -2, ...].
%F A134013 Moebius transform is period 16 sequence [ 1, 1, -1, -2, 1, -1, -1, 0, 
               1, 1, -1, 2, 1, -1, -1, 0, ...].
%F A134013 Multiplicative with a(2) = 2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 
               1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).
%F A134013 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 
               (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A134014.
%F A134013 a(4*n) = a(4*n+3) = a(8*n+6) = 0. a(8*n+2) = 2 * a(4*n+1).
%F A134013 G.f.: Sum_{k>0} kronecker(-4, k) * x^k * (1 + x^k)^2 / (1 - x^(4*k)).
%e A134013 q + 2*q^2 + 2*q^5 + q^9 + 4*q^10 + 2*q^13 + 2*q^17 + 2*q^18 + 3*q^25 
               + ...
%o A134013 (PARI) {a(n) = if( n>0 & (n+1)%4\2, (n%4) * sumdiv( n/gcd(n,2), d, (-1)^(d\2)))}
%o A134013 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 
               + A)^5 * eta(x^16 + A)^2 / eta(x + A)^2 / eta(x^4 + A)^2 / eta(x^8 
               + A), n))}
%Y A134013 Cf. A112301(n) = -(-1)^n * a(n). A008441(n) = a(4*n+1). A113407(n) = 
               a(8*n+1). 2 * A053692(n) = a(8*n+5).
%Y A134013 Sequence in context: A000095 A034949 A112301 this_sequence A136521 A066448 
               A108497
%Y A134013 Adjacent sequences: A134010 A134011 A134012 this_sequence A134014 A134015 
               A134016
%K A134013 nonn,mult
%O A134013 1,2
%A A134013 Michael Somos, Oct 02 2007

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