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%I #12 Mar 30 2012 18:37:29
%S 1,-3,-1,2,3,1,1,6,0,-3,5,2,1,-1,5,-8,5,10,-7,2,3,-9,-1,6,4,-3,8,2,-3,
%T 3,-11,2,13,-15,5,-2,-5,5,13,-8,-7,9,3,-2,18,-1,-7,-4,-14,-6,7,-4,-3,
%U 2,1,6,7,7,-9,18,-13,7,14,-12,1,-7,1,0,-3,-13,7,6,-3,-5,22,-16,3,-1,-11,2,8,-5,-15,6,1,-9,3,18,1,10,-13,8,9,3,-15,-2,-7,6,16,-4,1,1,3,-2,49,-7,-9,-6,-1,-9,-3,-20,-13,-11,-11,-22,12,25,7,0,-6,5,3,-2,-18,4,7,4,-1,-7,-5,-2,-15,3,32,2,15,11,-1,12,5,-23,3,-2,-17,1,10,4,7,16,13,34,-2,-31,-11,-12
%N a(n) = Sum_{k=0..2*n} (-1)^k * moebius(2*n-k+1) * moebius(k+1).
%C It is conjectured that all integers appear an infinite number of times.
%H Paul D. Hanna, <a href="/A195588/b195588.txt">Table of n, a(n) for n = 0..1001</a>
%F G.f. A(x) satisfies: A(x^2) = M(x)*M(-x) where M(x) = Sum_{n>=0} moebius(n+1)*x^n.
%F G.f. A(x) = exp( Sum_{n>=1} A195589(2*n)*x^n/n ), where A195589 is the unsigned logarithmic derivative of the Moebius function A008683.
%e G.f.: A(x) = 1 - 3*x - x^2 + 2*x^3 + 3*x^4 + x^5 + x^6 + 6*x^7 +...
%e where A(x^2) = M(x)*M(-x) and M(x) begins:
%e M(x) = 1 - x - x^2 - x^4 + x^5 - x^6 + x^9 - x^10 - x^12 + x^13 + x^14 - x^16 +...+ moebius(n+1)*x^n +...
%e log(A(x)) = -3*x - 11*x^2 - 30*x^3 - 83*x^4 - 243*x^5 - 710*x^6 - 2061*x^7 - 6099*x^8 +...+ -A195589(2*n)*x^n/n +...
%e Positions of zeros begin:
%e [8,67,119,161,167,206,207,243,260,263,271,331,339,350,371,407,543,803,...].
%e Positions of other values of a(n) begin:
%e +1: [0,5,6,12,54,64,66,84,88,100,101,145,202,210,256,290,309,318,321,...];
%e -1: [2,13,22,45,77,108,128,138,165,180,216,229,236,348,389,390,418,...];
%e +2: [3,11,19,27,31,53,79,135,242,360,362,413,548,800,839,...];
%e -2: [35,43,95,103,123,131,143,152,159,197,235,251,299,324,337,349,...];
%e +3: [4,20,29,42,76,86,93,102,122,133,142,201,240,326,333,401,518,585,...];
%e -3: [1,9,25,28,52,68,72,110,166,196,204,234,253,280,340,432,472,653,...];
%e +4: [24,125,127,147,170,211,269,278,332,459,807,...];
%e -4: [47,51,99,168,422,538,599,...];
%e +5: [10,14,16,34,37,121,140,177,308,382,484,520,537,642,645,706,741,...];
%e -5: [36,73,81,130,173,186,193,217,232,257,302,312,357,373,444,448,...].
%o (PARI) {a(n)=sum(k=0,2*n,(-1)^k*moebius(2*n-k+1)*moebius(k+1))}
%o (PARI) {A195589(n)=n*polcoeff(-log(sum(m=0,n,moebius(m+1)*x^m)+x*O(x^n)),n)}
%o {a(n)=polcoeff(exp(sum(m=1,n,-A195589(2*m)*x^m/m)+x*O(x^n)),n)}
%Y Cf. A195589, A008683 (Moebius).
%K sign
%O 0,2
%A _Paul D. Hanna_, Sep 20 2011
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