Search: id:A133985
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%I A133985
%S A133985 1,1,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,
%T A133985 1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
%U A133985 1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0
%V A133985 1,-1,1,0,0,-1,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,
               0,0,-1,0,0,0,0,
%W A133985 1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
               0,0,0,-1,0,0,0,
%X A133985 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0
%N A133985 Expansion of phi(q^3) / chi(q) in powers of q where phi(), chi() are 
               Ramanujan theta functions.
%F A133985 Expansion of f(-q, q^2) in powers of q where f() is the Ramanujan two 
               variable theta function.
%F A133985 Expansion of q^(-1/24) * eta(q) * eta(q^4) * eta(q^6)^5 / ( eta(q^2) 
               * eta(q^3) * eta(q^12) )^2 in powers of q.
%F A133985 Euler transform of period 12 sequence [ -1, 1, 1, 0, -1, -2, -1, 0, 1, 
               1, -1, -1, ...].
%F A133985 a(n) = b(24*n+1) where b(n) is multiplicative with b(p^(2e)) = (-1)^e 
               if p == 3, 5 (mod 8), b(p^(2e)) = +1 if p == 1, 7 (mod 8) and b(p^(2e-1)) 
               = b(2^e) = b(3^e) = 0 if e>0.
%F A133985 G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 
               4 (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A133988.
%F A133985 a(5n+3) = a(5n+4) = 0. a(25n+1) = -a(n).
%F A133985 G.f. Sum_{k>=0} a(k) x^(24k+1) = Sum_{k} (-1)^[k/2] x^(6k+1)^2.
%e A133985 q - q^25 + q^49 - q^121 - q^169 + q^289 - q^361 + q^529 + q^625 + ...
%o A133985 (PARI) {a(n) = (-1)^n * issquare( 24*n+1) }
%o A133985 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x 
               + A) * eta(x^4 + A) * eta(x^6 + A)^5 / ( eta(x^2 + A) * eta(x^3 + 
               A) * eta(x^12 + A) )^2, n))}
%Y A133985 (-1)^n * A080995(n) = a(n).
%Y A133985 Sequence in context: A010815 A080995 A121373 this_sequence A143062 A074910 
               A115356
%Y A133985 Adjacent sequences: A133982 A133983 A133984 this_sequence A133986 A133987 
               A133988
%K A133985 sign
%O A133985 0,1
%A A133985 Michael Somos, Oct 01 2007, Oct 04 2007

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