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A245735
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G.f. A(x) satisfies 0 = A(0) and 0 = f(x, A(x)) where f(u, v) = (u - v) * (1 + u*v) - u*v * (1 - u*v).
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2
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0, 1, -1, 1, 1, -5, 9, -3, -29, 89, -107, -121, 833, -1703, 631, 6705, -21943, 27587, 34937, -242427, 507739, -172615, -2201619, 7253775, -9083263, -12931023, 86757487, -181330015, 52436881, 843605643, -2751023447, 3373429837, 5393254483, -34585122919
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OFFSET
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0,6
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LINKS
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FORMULA
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G.f.: (-1 - x + x^2 + sqrt(1 + 2*x + 3*x^2 - 6*x^3 + x^4)) / (2 * (x - x^2)).
G.f.: x / (1 + x - x^2 + (x - x^2) * x / (1 + x - x^2 + (x - x^2) * x / ...)) continued fraction.
0 = +a(n)*(n) +a(n+1)*(-7*n-9) +a(n+2)*(9*n+18) +a(n+3)*(-n) +a(n+4)*(-n-3) +a(n+5)*(-n-6) for all n in Z.
0 = a(n)*(+a(n+1) -16*a(n+2) +27*a(n+3) -a(n+4) -4*a(n+5) -7*a(n+6)) + a(n+1)*(+2*a(n+1) +40*a(n+2) -109*a(n+3) +24*a(n+5) +40*a(n+6)) + a(n+2)*(-18*a(n+2) +97*a(n+3) +4*a(n+4) -44*a(n+5) -45*a(n+6)) +a(n+3)*(-27*a(n+3) +a(n+4) +31*a(n+5) +7*a(n+6)) +a(n+4)*(-2*a(n+4) -4*a(n+5) +4*a(n+6)) +a(n+5)*(-2*a(n+5) +a(n+6)) for all n in Z.
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EXAMPLE
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G.f. = x - x^2 + x^3 + x^4 - 5*x^5 + 9*x^6 - 3*x^7 - 29*x^8 + 89*x^9 + ...
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MATHEMATICA
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CoefficientList[Series[(-1 - x + x^2 + Sqrt[1 + 2*x + 3*x^2 - 6*x^3 + x^4])/(2*(x - x^2)), {x, 0, 60}], x] (* G. C. Greubel, Aug 06 2018 *)
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PROG
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(PARI) {a(n) = my(A); A = O(x); if( n<0, for(k=1, -n, A = x / (1 - (x + x^2) - (1 - x) * A)), for(k=0, n/2, A = x / (1 + (x - x^2) + (x - x^2) * A))); polcoeff(A, abs(n)) };
(PARI) {a(n) = polcoeff( if( n<0, ((1 - x - x^2) - sqrt(1 - 6*x + 3*x^2 + 2*x^3 + x^4 + x * O(x^-n) )) / (2 * (1 - x)), ((-1 - x + x^2) + sqrt(1 + 2*x + 3*x^2 - 6*x^3 + x^4 + x^2 * O(x^n) )) / (2 * (x - x^2))), abs(n))};
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((-1 - x +x^2 +Sqrt(1+2*x+3*x^2-6*x^3+x^4))/(2*(x-x^2)))); // G. C. Greubel, Aug 06 2018
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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