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%I #66 Oct 20 2012 03:00:32
%S 1,6,15,20,15,0,-29,-60,-66,-60,-51,0,60,84,135,160,81,60,70,-60,-120,
%T -108,-255,-240,-50,-126,-120,100,30,60,331,180,90,360,195,0,120,-180,
%U -306,80,-72,-312,-120,-300,-360,60,-51,-240,171,300,150,260,135,-60,240,420
%N G.f.: [ Sum_{n>=0} (-1)^n * x^(n*(3*n+2)) * (1 + x^(2*n+1)) ]^6.
%C The number of zero terms is infinite.
%H Paul D. Hanna, <a href="/A217480/b217480.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f. A(x) satisfies:
%F A(x^2)^(1/6) = Sum_{n>=0} x^n / Product_{k=0..n} (1 + x^(2*k+1)), by an identity due to Ramanujan.
%e G.f.: A(x) = 1 + 6*x + 15*x^2 + 20*x^3 + 15*x^4 - 29*x^6 - 60*x^7 - 66*x^8 +...
%e By definition, A(x) = B(x)^6 where
%e B(x) = 1 + x - x^5 - x^8 + x^16 + x^21 - x^33 - x^40 + x^56 + x^65 - x^85 - x^96 + x^120 + x^133 - x^161 - x^176 + x^208 + x^225 +...
%e By Ramanujan's identity,
%e B(x^2) = 1/(1+x) + x/((1+x)*(1+x^3)) + x^2/((1+x)*(1+x^3)*(1+x^5)) + x^3/((1+x)*(1+x^3)*(1+x^5)*(1+x^7)) + x^4/((1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)) +...
%e Zero-valued coefficients in g.f. A(x) = B(x)^6 are found at positions:
%e [5, 11, 35, 69, 75, 151, 177, 180, 226, 516, 539, 728, 922, 2544, 3320, 3936, 4796, 5658, 5702, 6968, 7887, 8603, 9472, ...];
%e higher powers of B(x) have at most a finite number of zero coefficients.
%o (PARI) {a(n)=local(A=sum(m=0,n,(-1)^m*x^(m*(3*m+2))*(1+x^(2*m+1))+x*O(x^n)));polcoeff(A^6,n)}
%o for(n=0,60,print1(a(n),", "))
%K sign
%O 0,2
%A _Paul D. Hanna_, Oct 20 2012
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