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A302699
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G.f. A(x) satisfies: A(x) = 1 + x * A(x) * (A(x) + 3*x*A'(x)) / (A(x) + x*A'(x)).
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1
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1, 1, 3, 13, 71, 469, 3711, 35181, 398791, 5352149, 83650687, 1494274301, 29988083447, 666634964197, 16233361360559, 429237520044813, 12237655701598503, 374023408217062261, 12195222470567359071, 422440153967133458205, 15490152522612488256855, 599350023954941335582725, 24401304036660493806643215
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OFFSET
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0,3
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COMMENTS
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Compare to: C(x) = 1 + x*C(x) * (C(x) + 2*x*C'(x)) / (C(x) + x*C'(x)) holds when C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = 1 + x*A(x) * (A(x) + 3*x*A'(x)) / (A(x) + x*A'(x)).
(2) A(x) = 1/(1 - x*A(x)/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - 5*x*A(x)/(1 - ...)))))), a continued fraction.
(3) A(x) = Series_Reversion( x - x^2*F(x) ) where F(x) = Sum_{n>=0} (2*n)!/(n!*2^n)*x^n (g.f. of the odd double factorials A001147).
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 71*x^4 + 469*x^5 + 3711*x^6 + 35181*x^7 + 398791*x^8 + 5352149*x^9 + 83650687*x^10 + ...
RELATED SERIES.
A'(x)/A(x) = 1 + 5*x + 31*x^2 + 225*x^3 + 1891*x^4 + 18473*x^5 + 210939*x^6 + 2815137*x^7 + 43551715*x^8 + 770297385*x^9 + ...
A(x) + x*A'(x) = 1 + 2*x + 9*x^2 + 52*x^3 + 355*x^4 + 2814*x^5 + 25977*x^6 + 281448*x^7 + 3589119*x^8 + 53521490*x^9 + ...
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PROG
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(PARI) /* Differential equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A*(A + 3*x*A')/(x*A +x^2*O(x^n))'); polcoeff(G=A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, for(k=0, n, CF = 1/(1 - (n-k+1)*x*A*CF ) ); A=1/(1 - x*A*CF) ); polcoeff(A, n)}
for(n=0, 30, print1(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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