Search: id:A096727
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%I A096727
%S A096727 1,8,24,32,24,48,96,64,24,104,144,96,96,112,192,192,24,144,312,160,144,
%T A096727 256,288,192,96,248,336,320,192,240,576,256,24,384,432,384,312,304,480,
%U A096727 448,144,336,768,352,288,624,576,384,96,456,744,576,336,432,960,576,192
%V A096727 1,-8,24,-32,24,-48,96,-64,24,-104,144,-96,96,-112,192,-192,24,-144,312,
               -160,144,-256,
%W A096727 288,-192,96,-248,336,-320,192,-240,576,-256,24,-384,432,-384,312,-304,
               480,-448,144,
%X A096727 -336,768,-352,288,-624,576,-384,96,-456,744,-576,336,-432,960,-576,192
%N A096727 Expansion of eta(q)^8/eta(q^2)^4 in powers of q.
%C A096727 Euler transform of period 2 sequence [ -8,-4,...].
%F A096727 G.f. Prod_{k>0} (1-x^k)^8/(1-x^(2k))^4 = 1 +Sum_{k>0} k(-8x^k/(1-x^k) 
               +48x^(2k)/(1-x^(2k))-64x^(4k)/(1-x^(4k))).
%F A096727 G.f. theta_4(q)^4 = (Sum_{k} (-q)^(k^2))^4.
%F A096727 Expansion of phi(-q)^4 in powers of q where phi() is a Ramanujan theta 
               function. - Michael Somos Nov 01 2006
%F A096727 G.f. A(x) satisfies 0=f(A(x), A(x^3), A(x^9)) where f(u, v, w) = v^4 
               -30*u*v^2*w +12*u*v*w*(u +9*w) -u*w*(u^2 +9*w*u +81*w^2).
%t A096727 CoefficientList[ Series[1 + Sum[k(-8x^k/(1 - x^k) + 48x^(2k)/(1 - x^(2k)) 
               - 64x^(4k)/(1 - x^(4k))), {k, 1, 60}], {x, 0, 60}], x] (from Robert 
               G. Wilson v Jul 14 2004)
%o A096727 (PARI) a(n)=if(n<1,n==0,8*(-1)^n*sumdiv(n,d,if(d%4,d)))
%o A096727 (PARI) a(n)=local(A); if(n<0,0,A=x*O(x^n); polcoeff(eta(x+A)^8/eta(x^2+A)^4,
               n))
%Y A096727 A000118(n)=(-1)^n*a(n). A109506(n)=a(n)/8 if n>0. A004011(n)=a(2n). A005879(n)=-a(2n+1).
%Y A096727 Sequence in context: A038524 A162829 A000118 this_sequence A028660 A028644 
               A056196
%Y A096727 Adjacent sequences: A096724 A096725 A096726 this_sequence A096728 A096729 
               A096730
%K A096727 sign
%O A096727 0,2
%A A096727 Michael Somos, Jul 06 2004

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