A191565 G.f. satisfies: A(A(x))^2 = A(x)^2 + 4*x^3.
1, 2, -14, 184, -3194, 65472, -1503924, 37593664, -1004163802, 28314667072, -835650200380, 25652840146624, -815280469973380, 26728163562423360, -901336722528156712, 31194183364269262848, -1105930698812430437626
Offset: 1
Keywords
Examples
G.f.: A(x) = x + 2*x^2 - 14*x^3 + 184*x^4 - 3194*x^5 + 65472*x^6 +... Illustrate A(A(x))^2 - A(x)^2 = 4*x^3 with the expansions: A(x)^2 = x^2 + 4*x^3 - 24*x^4 + 312*x^5 - 5456*x^6 + 113016*x^7 +... A(A(x))^2 = x^2 + 8*x^3 - 24*x^4 + 312*x^5 - 5456*x^6 + 113016*x^7 +... A(A(x)) = x + 4*x^2 - 20*x^3 + 236*x^4 - 3872*x^5 + 76716*x^6 - 1723488*x^7 +... Let R(x) be the series reversion of A(x): R(x) = x - 2*x^2 + 22*x^3 - 364*x^4 + 7390*x^5 - 170556*x^6 +... R(x)^3 = x^3 - 6*x^4 + 78*x^5 - 1364*x^6 + 28254*x^7 - 655668*x^8 +... where A(x)^2 = x^2 + 4*R(x)^3.
Programs
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PARI
{a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=A-(subst(A,x,A)-x*sqrt(4*x+A^2/x^2)));polcoeff(A,n)}