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G.f.: 1/(1-x) + Sum_{n>=1} x^n/(1-x) * Product_{k=1..n} (1 - x^k) / (1 - x^(2*k+1)).
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%I #20 Mar 30 2012 18:37:33

%S 1,2,2,2,2,2,1,2,2,0,2,2,0,2,2,0,2,2,1,2,2,0,2,4,0,2,2,0,4,2,0,2,2,2,

%T 2,2,-1,2,4,0,2,2,0,2,2,0,2,2,2,2,0,0,2,4,0,2,2,-2,4,2,1,2,2,2,0,2,0,

%U 2,4,2,2,0,0,4,2,0,2,2,2,2,2,0,2,4,0,0

%N G.f.: 1/(1-x) + Sum_{n>=1} x^n/(1-x) * Product_{k=1..n} (1 - x^k) / (1 - x^(2*k+1)).

%H Paul D. Hanna, <a href="/A202146/b202146.txt">Table of n, a(n) for n = 0..2106</a>

%F a(k) == 1 (mod 2) iff k = 3*n*(n+1) for n>=0 (conjecture).

%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + x^6 + 2*x^7 +...

%e where A(x) = 1/(1-x) + x*(1-x)/((1-x)*(1-x^3)) + x^2*(1-x)*(1-x^2)/((1-x)*(1-x^3)*(1-x^5)) + x^3*(1-x)*(1-x^2)*(1-x^3)/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)) +...

%e which is a q-series analog of the sum:

%e Pi/2 = 0!/1 + 1!/(1*3) + 2!/(1*3*5) + 3!/(1*3*5*7) + 4!/(1*3*5*7*9) + 5!/(1*3*5*7*9*11) +...

%e Odd terms (A202150), located at positions 3*n*(n+1) for n>=0, begin:

%e [1,1,1,-1,1,-1,1,3,1,-1,1,-1,1,1,1,-1,-1,1,1,1,1,-1,3,-1,1,1,1,...].

%o (PARI) {a(n)=polcoeff((1+sum(m=1,n,x^m*prod(k=1,m,(1-x^k)/(1-x^(2*k+1) +x*O(x^n)))))/(1-x+x*O(x^n)),n)}

%Y Cf. A202145 (first differences), A202150 (odd terms).

%K sign

%O 0,2

%A _Paul D. Hanna_, Dec 12 2011