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A087949
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G.f. satisfies A(x) = 1 + xA(xA(x)).
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10
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1, 1, 1, 2, 5, 16, 59, 246, 1131, 5655, 30428, 174835, 1066334, 6870542, 46581883, 331237074, 2463361903, 19112314727, 154364077009, 1295325828045, 11273167827343, 101589943242179, 946577526626181, 9107029927925714
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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FORMULA
| Let G(x) = x*A(x), then the following statements hold:
* G(x) = x*(1 + sqrt(1 + 4*G(G(x))))/2;
* G(x) = Series_Reversion[2*x/(1 + sqrt(1 + 4*G(x)))].
- Paul D. Hanna (pauldhanna(AT)juno.com), May 15 2008
Comment from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007: G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + xB;
B = 1 + xAC;
C = 1 + xABD;
D = 1 + xABCE;
E = 1 + xABCDF ; ...
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009: (Start)
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n, then
a(n,m) = Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * a(n-k,k) with a(0,m)=1.
(End)
G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(n+1)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
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EXAMPLE
| G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 +...
A(xA(x)) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 59*x^5 +...
Logarithmic series:
log(A(x)) = x/A(x) + [d/dx x^3*A(x)^2]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^3]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^4]*A(x)^(-8)/4! +...
Let G(x) = x*A(x) then
x = G(x*[1 - G(x) + 2*G(x)^2 - 5*G(x)^3 + 14*G(x)^4 - 42*G(x)^5 +-...])
where the unsigned coefficients are the Catalan numbers (A000108).
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PROG
| (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(2*x/(1 + sqrt(1+4*A +x*O(x^n))))); polcoeff(A, n))}
(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n-k+m, k)/(n-k+m)*a(n-k, k))))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009]
(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^(m+1))*A^(-2*m-2)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
| Cf. A002449, A030266, A088714, A088717, A091713, A120971, A140092, A000108.
Cf. A139702, A143426, A143435, A182969.
Sequence in context: A000753 A007878 A019589 * A028333 A007747 A107283
Adjacent sequences: A087946 A087947 A087948 * A087950 A087951 A087952
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2003
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), May 19 2008
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