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A143340
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G.f. satisfies: A(x) = 1 + x*A(x)^3/A(-x).
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2
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1, 1, 4, 15, 84, 402, 2520, 13339, 88484, 494814, 3395816, 19657398, 137999048, 818024484, 5836517808, 35201610387, 254231733188, 1553691459558, 11327637588552, 69948932919906, 513856752260184, 3199802098978428
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| G.f. satisfies: A(x) - x*A(x)^3 = 1 + x^2*[A(x)*A(-x)]^2.
G.f. satisfies:
_ A(x) = Sum_{n>=1} x^n * A(x)^(2*n)/A(-x)^n ;
_ A(x) = exp( Sum_{n>=1} x^n/n * A(x)^(2*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)^3:
A(x) = 1 + x + 4*x^2 + 15*x^3 + 84*x^4 + 402*x^5 + 2520*x^6 + 13339*x^7 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 70*x^3 + 402*x^4 + 2163*x^5 + 13339*x^6 +...
so that A(x) - x*A(x)^3 = 1 + x^2*[A(x)*A(-x)]^2, where
[A(x)*A(-x)]^2 = 1 + 14*x^2 + 357*x^4 + 11522*x^6 + 420170*x^8 +...
A(x)*A(-x) = 1 + 7*x^2 + 154*x^4 + 4683*x^6 + 165446*x^8 +...
Related expressions.
A(x) = 1 + x*A(x)^2/A(-x) + x^2*A(x)^4/A(-x)^2 + x^3*A(x)^6/A(-x)^3 +...
log(A(x)) = x*A(x)^2/A(-x) + x^2/2*A(x)^4/A(-x)^2*x^2 + x^3/3*A(x)^6/A(-x)^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^3/subst(A, x, -x)); 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^(2*m)/subst(A^m, x, -x+x*O(x^n)))); 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^(2*m)*subst(A^-m, x, -x)*x^m/m)+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
| Cf. A143339, A143341.
Sequence in context: A129653 A081722 A117927 * A151379 A130679 A107874
Adjacent sequences: A143337 A143338 A143339 * A143341 A143342 A143343
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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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