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A143339
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G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x).
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3
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1, 1, 3, 7, 25, 73, 283, 911, 3697, 12561, 52467, 184471, 785929, 2829401, 12229259, 44795167, 195742177, 726541345, 3202144483, 12010174247, 53300753657, 201608659561, 899838791419, 3427434566831, 15370709035601, 58890032580913
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| G.f. satisfies: 1 - 2*A(x) + (1+x)*A(x)^2 - (x+x^3)*A(x)^3 = 0.
G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x)^n*A(-x)^-n * x^n/n ). [From Paul D. Hanna (pauldhanna(AT)juno.com), Dec 04 2009]
G.f. satisfies: A(x) = Sum_{n>=1} x^n * A(x)^n / A(-x)^n. [From Paul D. Hanna, Sep 30 2011]
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EXAMPLE
| A bisection of g.f. A(x) equals a bisection of A(x)^2:
A(x) = 1 + x + 3*x^2 + 7*x^3 + 25*x^4 + 73*x^5 + 283*x^6 + 911*x^7 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 20*x^3 + 73*x^4 + 238*x^5 + 911*x^6 +...
Also, A(x) - x*A(x)^2 = 1 + x^2*A(x)*A(-x), where
A(x)*A(-x) = 1 + 5*x^2 + 45*x^4 + 521*x^6 + 6873*x^8 + 98061*x^10 +...
Related expressions.
A(x) = 1 + x*A(x)/A(-x) + x^2*A(x)^2/A(-x)^2 + x^3*A(x)^3/A(-x)^3 +...
log(A(x)) = A(x)/A(-x)*x + A(x)^2/A(-x)^2*x^2/2 + A(x)^3/A(-x)^3*x^3/3 +...
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PROG
| (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*A^2/subst(A, x, -x)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, A^m/subst(A^m, x, -x+x*O(x^n))*x^m/m))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, x^m*A^m/subst(A^m, x, -x+x*O(x^n)))); polcoeff(A, n)}
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CROSSREFS
| Sequence in context: A148727 A148728 A148729 * A148730 A148731 A148732
Adjacent sequences: A143336 A143337 A143338 * A143340 A143341 A143342
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 09 2008
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