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A143435
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G.f. satisfies: A(x) = 1 + x*A(x*A(x))^3.
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6
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1, 1, 3, 15, 97, 738, 6297, 58630, 585543, 6200916, 69071103, 804470751, 9753459717, 122670681073, 1596129692136, 21437840848440, 296680980737270, 4224090724829151, 61794432127467450, 927795254532531834
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| G.f. satisfies: x - G(x) = G(x)^2*A(x)^3 where G(x*A) = x.
G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(3*n+3)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
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EXAMPLE
| G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 97*x^4 + 738*x^5 + 6297*x^6 +...
A(x*A(x)) = 1 + x + 4*x^2 + 24*x^3 + 178*x^4 + 1511*x^5 + 14130*x^6 +...
A(x*A(x))^3 = 1 + 3*x + 15*x^2 + 97*x^3 + 738*x^4 + 6297*x^5 +...
Logarithmic series:
log(A(x)) = x*A(x) + [d/dx x^3*A(x)^6]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^9]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^12]*A(x)^(-8)/4! +...
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PROG
| (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, 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^(3*m+3))*A^(-2*m-2)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
| Cf. A143426, A143436, A143437.
Cf. A139702, A087949, A182969.
Sequence in context: A079689 A108442 A060148 * A132437 A128081 A186264
Adjacent sequences: A143432 A143433 A143434 * A143436 A143437 A143438
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 14 2008
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