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A273958
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G.f. A(x) satisfies: x*A(x) + x^2*A(x)^2 = C(x)^2, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
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2
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1, 2, 4, 10, 30, 96, 313, 1038, 3512, 12100, 42286, 149440, 533076, 1917000, 6942660, 25300034, 92701746, 341319648, 1262176462, 4685724500, 17457027376, 65247128392, 244584468174, 919319871360, 3464027857900, 13082491517336, 49513139090512, 187761168576904, 713325243149880, 2714644395230080, 10347465825694361, 39500936075084958, 151005816167189208, 578038682649831252, 2215461387843704450, 8501300598574467776, 32658238309503587964
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OFFSET
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1,2
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COMMENTS
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It appears that a(n) = 1 (mod 2) iff n = 2*4^k - 1 for k>=0.
Radius of convergence is 1/4 with A(1/4) = 2*(sqrt(2) - 1).
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = (sqrt(1 + 4*C(x)^2) - 1)/(2*x), where C(x) = (1 - sqrt(1 - 4*x))/2.
(2) A(x) = (1/x) * Series_Reversion( sqrt(x^2 + x^4) - (x^2 + x^4) )^2.
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EXAMPLE
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G.f.: A(x) = x + 2*x^2 + 4*x^3 + 10*x^4 + 30*x^5 + 96*x^6 + 313*x^7 + 1038*x^8 + 3512*x^9 + 12100*x^10 + 42286*x^11 + 149440*x^12 + +...
such that x*A(x) + x^2*A(x)^2 = C(x)^2, where
A(x)^2 = x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 116*x^6 + 392*x^7 + 1350*x^8 + 4696*x^9 + 16500*x^10 + 58572*x^11 + 209824*x^12 +...+ A055395(n+2)*x^n +...
C(x)^2 = x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 +...+ A000108(n)*x^n +...
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PROG
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(PARI) /* From A(x) = (sqrt(1 + 4*C(x)^2) - 1)/(2*x) */
{a(n) = my(A=x, C=(1 - sqrt(1 - 4*x +x*O(x^n)))/2); A = (sqrt(1 + 4*C^2) - 1)/(2*x); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
(PARI) /* From A(x) = (1/x)*Series_Reversion( sqrt(x^2 + x^4) - (x^2 + x^4) )^2 */
{a(n) = my(A=x); A = ( serreverse( sqrt(x^2 + x^4 +x^3*O(x^n)) - (x^2 + x^4) ) )^2/x; polcoeff(A, n)}
for(n=1, 40, 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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