Search: id:A132973
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%I A132973
%S A132973 1,3,3,3,3,0,3,6,3,3,0,0,3,6,6,0,3,0,3,6,0,6,0,0,3,3,6,3,6,0,0,6,3,0,0,
%T A132973 0,3,6,6,6,0,0,6,6,0,0,0,0,3,9,3,0,6,0,3,0,6,6,0,0,0,6,6,6,3,0,0,6,0,0,
%U A132973 0,0,3,6,6,3,6,0,6,6,0,3,0,0,6,0,6,0,0,0,0,12,0,6,0,0,3,6,9,0,3,0,0,6
%V A132973 1,-3,3,-3,3,0,3,-6,3,-3,0,0,3,-6,6,0,3,0,3,-6,0,-6,0,0,3,-3,6,-3,6,0,
               0,-6,3,0,0,0,3,
%W A132973 -6,6,-6,0,0,6,-6,0,0,0,0,3,-9,3,0,6,0,3,0,6,-6,0,0,0,-6,6,-6,3,0,0,-6,
               0,0,0,0,3,-6,6,
%X A132973 -3,6,0,6,-6,0,-3,0,0,6,0,6,0,0,0,0,-12,0,-6,0,0,3,-6,9,0,3,0,0,-6
%N A132973 Expansion of psi(-q)^3 / psi(-q^3) in powers of q where psi() is a Ramanujan 
               theta function.
%F A132973 Expansion of b(q^2)^2 / b(-q) in powers of q where b() is a cubic AGM 
               function.
%F A132973 Expansion of eta(q)^3 * eta(q^4)^3 * eta(q^6) / ( eta(q^2)^3 * eta(q^3) 
               * eta(q^12) ) in powers of q.
%F A132973 Euler transform of period 12 sequence [ -3, 0, -2, -3, -3, 0, -3, -3, 
               -2, 0, -3, -2, ...].
%F A132973 Moebius transform is period 12 sequence [ -3, 6, 0, 0, 3, 0, -3, 0, 0, 
               -6, 3, 0, ...].
%F A132973 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108^(1/
               2) (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A113447.
%F A132973 a(6*n+5) = 0.
%F A132973 G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^3 / ((1 - x^(3*k)) * 
               (1 + x^(6*k))).
%F A132973 G.f.: 1 + 3 * Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
%F A132973 G.f.: 1 + 3 * ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / 
               ( 1 + x^(6*k-1) )).
%e A132973 1 - 3*q + 3*q^2 - 3*q^3 + 3*q^4 + 3*q^6 - 6*q^7 + 3*q^8 - 3*q^9 + 3*q^12 
               + ...
%o A132973 (PARI) {a(n) = if( n<1, n==0, 3 * (-1)^n * sumdiv(n, d, kronecker(-12, 
               d)))}
%o A132973 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x 
               + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) / ( eta(x^2 + A)^3 * eta(x^3 
               + A) * eta(x^12 + A ) ), n))}
%Y A132973 (-1)^n * A107760(n) = a(n). Convolution inverse of A132974.
%Y A132973 Sequence in context: A031354 A033700 A122916 this_sequence A107760 A138070 
               A081334
%Y A132973 Adjacent sequences: A132970 A132971 A132972 this_sequence A132974 A132975 
               A132976
%K A132973 sign
%O A132973 0,2
%A A132973 Michael Somos, Sep 07 2007

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