A378261 - OEIS

%I #15 Nov 27 2024 10:04:50

%S 1,-1,-2,1,-2,3,3,-2,0,1,5,-3,1,-6,-2,5,-5,-1,-1,1,10,-3,-18,4,0,6,-2,

%T 2,-7,3,10,-8,3,13,3,-4,-5,-1,16,-7,1,-21,-21,13,11,32,-3,-18,14,-12,

%U 27,-5,-29,-14,40,18,-9,-7,-28,-9,28,-12,-1,21,-18,-17,21,-4,2,-26,-55,15,6,57,76,4,-63,-35,-31,12,27

%N G.f. satisfies A(x) = A(x^2)*M(x)/x, where M(x) = Sum_{n>=1} mu(n)*x^n and mu(n) = A008683(n), the Moebius function of n.

%C Given g.f. A(x), x/A(x) equals the g.f. of A378260.

%H Paul D. Hanna, <a href="/A378261/b378261.txt">Table of n, a(n) for n = 0..8200</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A(x) = A(x^2)*M(x)/x, where M(x) = Sum_{n>=1} mu(n)*x^n.

%F (2) x = Sum_{n>=1} x^n * A(x^n)/A(x^(2*n)).

%e G.f.: A(x) = 1 - x - 2*x^2 + x^3 - 2*x^4 + 3*x^5 + 3*x^6 - 2*x^7 + x^9 + 5*x^10 - 3*x^11 + x^12 - 6*x^13 - 2*x^14 + 5*x^15 - 5*x^16 - x^17 - x^18 + x^19 + 10*x^20 - 3*x^21 - 18*x^22 + 4*x^23 + 6*x^25 + ...

%e where A(x) = A(x^2)*M(x)/x, with M(x) starting as

%e M(x) = x - x^2 - x^3 - x^5 + x^6 - x^7 + x^10 - x^11 - x^13 + x^14 + x^15 - x^17 - x^19 + x^21 + x^22 - x^23 + x^26 - x^29 - x^30 + ... + mu(n)*x^n + ...

%e so that x = M(x) + M(x^2) + M(x^3) + M(x^4) + ... + M(x^n) + ...

%e Thus, because M(x) = x*A(x)/A(x^2), we have

%e x = x*A(x)/A(x^2) + x^2*A(x^2)/A(x^4) + x^3*A(x^3)/A(x^6) + x^4*A(x^4)/A(x^8) + x^5*A(x^5)/A(x^10) + ... + x^n*A(x^n)/A(x^(2*n)) + ...

%e SPECIFIC VALUES.

%e A(z) = 0 at z = 0.58029462380732672306477623722678043664946494150129261513733196007593...

%e   at which M(z) = 0; also,

%e   A(+/-sqrt(z)) = 0 where sqrt(z) =  0.76177071603424525645824609227751503049205504631631398431010255922079...

%e A(t) = 1 at t = -0.36788404155611343004262030413356489215751069093739780571442285347...

%e   and at t = 0.

%e A(t) = 3/4 at t = 0.1857761904825755826296582919924203148981077141603702423677032...

%e   and at t = -0.490458121286701964509344440548675674917868267307919314675917...

%e A(t) = 2/3 at t = 0.2335393155019759242193743786104849997659161888659795344146053...

%e   and at t = -0.519733956808611007765606585829046829235954761454838631891973...

%e A(t) = 1/2 at t = 0.3198405467887820951152560564404574186606511299096542831871690...

%e   and at t = -0.571809669500110081208798541641929396382408973747748977426161...

%e A(t) = 1/3 at t = 0.3999205691655568373317765145229363880031158016542362401450436...

%e   and at t = -0.620228052536454037811510479582708461188023489655490588138156...

%e A(t) = 1/4 at t = 0.4397125538469644065749981128335829295493172812091133616367549...

%e   and at t = -0.644814296439482673338297235928747677433880338824104561837150...

%e A(2/3) = -0.061742613282000323935088569460595490025354693881260621358878068...

%e A(1/2) = 0.1298319421606939473389436822721728169327823265614184406479845097...

%e A(1/3) = 0.4724530047506755119306074032801526676340305286513455974985300628...

%e A(1/4) = 0.6363603730534968288384199665118751095126750453913421545923659477...

%e A(1/9) = 0.8653204774591832015787933929093873031717584973772619782252127819...

%e A(1/16) = 0.929904155452109003231349129681442625998610476992091204616469757...

%o (PARI) {a(n) = my(A=1-x, M = sum(m=1,n+1,moebius(m)*x^m) +x^2*O(x^n));

%o for(i=1,#binary(n)+1, A = subst(A,x,x^2)*M/x ); polcoef(A,n)}

%o for(n=0,80, print1(a(n),", "))

%Y Cf. A378260, A195589, A008683, A181439.

%K sign

%O 0,3

%A _Paul D. Hanna_, Nov 26 2024