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A143426
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G.f. satisfies: A(x) = 1 + x*A(x*A(x))^2.
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8
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1, 1, 2, 7, 32, 175, 1086, 7429, 54994, 435120, 3647686, 32192596, 297654824, 2872372828, 28841766844, 300592170551, 3244942353856, 36219458512421, 417365572999944, 4958429472475171, 60659660219655616, 763325035692109389
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. satisfies: x - G(x) = G(x)^2*A(x)^2 where G(x*A(x)) = x.
G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(2n+2)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 175*x^5 + 1086*x^6 +...
A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 434*x^5 + 2986*x^6 +...
A(x*A(x))^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 175*x^4 + 1086*x^5 +...
Logarithmic series:
log(A(x)) = x + [d/dx x^3*A(x)^4]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^6]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^8]*A(x)^(-8)/4! +...
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^2, x, x*A)); polcoeff(A, n)}
(PARI) /* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, Dec 18 2010] */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n,
A=exp(sum(m=0, n, Dx(m, x^(2*m+1)*A^(2*m+2))*A^(-2*m-2)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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