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A302705
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O.g.f. A(x) satisfies: A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.
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4
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1, 1, 3, 11, 43, 171, 677, 2637, 10035, 37171, 134009, 472785, 1655845, 5910373, 22254507, 90625475, 396822579, 1803795507, 8151776201, 35314777505, 142395796689, 518352934225, 1625522953935, 3944383216263, 4604242439037, -17114536692099, -114353748666873, -52384917067153, 4112292989447275, 42810794269242411, 290607272326013813
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OFFSET
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0,3
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COMMENTS
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Note: if F(x) = 1 + Integral (x*F(x)^3)' / (x*F(x))' dx, then F(x) does not consist entirely of integer coefficients in its power series expansion.
Given G(x) = 1 + Integral (x*G(x)^p)' / (x*G(x)^q)' dx, for what fixed integers p and q is G(x) an integer power series?
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LINKS
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FORMULA
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O.g.f. A(x) satisfies:
(1) A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.
(2) A(x) = 1 + Integral A(x)^3 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)) dx.
(3) A(x) = 1 + Integral A(x) * ( sqrt(1 - 4*x*A(x)^2 + 16*x^2*A(x)^4) - 1 + 4*x*A(x)^2 ) / (2*x) dx.
(4) 0 = A(x)^4 - A(x)*(1 - 4*x*A(x)^2)*A'(x) - x*A'(x)^2.
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EXAMPLE
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O.g.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 171*x^5 + 677*x^6 + 2637*x^7 + 10035*x^8 + 37171*x^9 + 134009*x^10 + 472785*x^11 + 1655845*x^12 + ...
RELATED SERIES.
(x*A(x)^4)' / (x*A(x))' = 1 + 6*x + 33*x^2 + 172*x^3 + 855*x^4 + 4062*x^5 + 18459*x^6 + 80280*x^7 + 334539*x^8 + ... + (n+1)*a(n+1)*x^n + ...
which equals A'(x).
The logarithmic derivative of the g.f. begins:
A'(x)/A(x) = 1 + 5*x + 25*x^2 + 121*x^3 + 561*x^4 + 2477*x^5 + 10361*x^6 + 40817*x^7 + 150433*x^8 + 515605*x^9 + 1646041*x^10 + ...
which equals (sqrt(1 - 4*x*A(x)^2 + 16*x^2*A(x)^4) - 1 + 4*x*A(x)^2) / (2*x).
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PROG
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(PARI) {a(n) = my(A=1); for(i=1, n, A = A = 1 + intformal( (x*A^4)'/(x*A +x*O(x^n))' ); ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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