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A139702
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G.f. satisfies: x = A( x + A(x)^2 ).
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12
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1, -1, 4, -24, 178, -1512, 14152, -142705, 1528212, -17211564, 202460400, -2474708496, 31310415376, -408815254832, 5495451727376, -75907303147652, 1075685334980240, -15618612118252960, 232102241507321384, -3526880759915999016
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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LINKS
| Paul D. Hanna, Table of n, a(n), n=1..100.
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FORMULA
| Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then G(x) = G(G(x)) - x^2 = g.f. of A138740.
G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 - xB^2;
B = A - xC^2;
C = B - xD^2;
D = C - xE^2;
E = D - xF^2; ...
G.f. satisfies: A(x) = x*exp( Sum_{n>=0} (-1)^(n+1)*[d^n/dx^n A(x)^(2n+2)/x]/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
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EXAMPLE
| G.f.: A(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
where A(x + A(x)^2) = x.
Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then:
G(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+... and
G(G(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+...
so that G(x) = G(G(x)) - x^2 = g.f. of A138740.
Logarithmic series:
log(A(x)/x) = -A(x)^2/x + [d/dx A(x)^4/x]/2! - [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! -+...
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PROG
| (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x + (A+x*O(x^n))^2)); 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=x-x^2+x*O(x^n)); for(i=1, n,
A=x*exp(sum(m=0, n, (-1)^(m+1)*Dx(m, A^(2*m+2)/x)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
| Cf. A138740.
Cf. A088714, A088717, A091713, A120971, A140094, A140095.
Cf. A143426, A087949, A143435, A182969.
Sequence in context: A000309 A112914 A007846 * A168452 A061720 A197472
Adjacent sequences: A139699 A139700 A139701 * A139703 A139704 A139705
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KEYWORD
| sign
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Apr 30 2008, May 20 2008
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