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A143341
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G.f. satisfies: A(x) = 1 + x*A(x)^4/A(-x).
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2
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1, 1, 5, 26, 195, 1303, 11076, 81910, 740151, 5782175, 54176573, 438029432, 4203769940, 34798104500, 339699218160, 2860590892318, 28283147265023, 241296800029199, 2409437282086511, 20767852798378330, 209017295575667771
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. satisfies: A(x) - x*A(x)^4 = 1 + x^2*[A(x)*A(-x)]^3.
G.f. satisfies:
_ A(x) = Sum_{n>=1} x^n * A(x)^(3*n)/A(-x)^n;
_ A(x) = exp( Sum_{n>=1} x^n/n * A(x)^(3*n)/A(-x)^n ). [From Paul D. Hanna, Sep 30 2011]
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EXAMPLE
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A bisection of g.f. A(x) equals a bisection of A(x)^4:
A(x) = 1 + x + 5*x^2 + 26*x^3 + 195*x^4 + 1303*x^5 + 11076*x^6 + 81910*x^7 +...
A(x)^4 = 1 + 4*x + 26*x^2 + 168*x^3 + 1303*x^4 + 9744*x^5 + 81910*x^6 +...
so that A(x) - x*A(x)^4 = 1 + x^2*[A(x)*A(-x)]^3, where
[A(x)*A(-x)]^3 = 1 + 27*x^2 + 1332*x^4 + 82791*x^6 + 5800329*x^8 +...
A(x)*A(-x) = 1 + 9*x^2 + 363*x^4 + 20820*x^6 + 1397511*x^8 +...
Related expressions.
A(x) = 1 + x*A(x)^3/A(-x) + x^2*A(x)^6/A(-x)^2 + x^3*A(x)^9/A(-x)^3 +...
log(A(x)) = x*A(x)^3/A(-x) + x^2/2*A(x)^6/A(-x)^2*x^2 + x^3/3*A(x)^9/A(-x)^3 +...
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*A^4/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^(3*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^(3*m)*subst(A^-m, x, -x)*x^m/m)+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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