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A199548
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G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 - A(x)^(2*n+1))/(1 - A(x)).
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2
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1, 3, 14, 88, 650, 5257, 45017, 401010, 3677344, 34481492, 329082191, 3186043296, 31214870385, 308901931412, 3083146893716, 31001118379636, 313734072027372, 3193097704841990, 32662597147529218, 335616736745247652, 3462524444288857191, 35853293611333010079
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OFFSET
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0,2
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COMMENTS
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Compare to g.f. B(x) of A007317 (binomial transform of Catalan numbers):
B(x) = Sum_{n>=0} x^n * (1 - B(x)^(n+1))/(1 - B(x)).
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LINKS
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FORMULA
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G.f. satisfies: A(x) = 1 + x*A(x)*(2 + A(x)^2) - x^2*A(x)^3.
G.f. satisfies: x = (2 + A(x)^2 - sqrt(4 + 4*A(x) + A(x)^4))/(2*A(x)^2).
D-finite with recurrence 2*n*(2*n+1)*a(n) +(-47*n^2+31*n-2)*a(n-1) +4*(2*n^2+53*n-88)*a(n-2) +(255*n^2-1693*n+2668)*a(n-3) +2*(-217*n^2+1622*n-3028)*a(n-4) +2*(139*n^2-1235*n+2746)*a(n-5) -32*(n-5)*(2*n-11)*a(n-6)=0. - R. J. Mathar, Feb 10 2024
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 14*x^2 + 88*x^3 + 650*x^4 + 5257*x^5 +...
where g.f. A = A(x) satisfies the equivalent expressions:
A = 1 + x*(1-A^3)/(1-A) + x^2*(1-A^5)/(1-A) + x^3*(1-A^7)/(1-A) +...
A = 1 + x*(1 + A + A^2) + x^2*(1 + A + A^2 + A^3 + A^4) + x^3*(1 + A + A^2 + A^3 + A^4 + A^5 + A^6) +...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, 2*m, A^k)+x*O(x^n))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A*(2+A^2)-x^2*A^3+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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