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A201969
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G.f. satisfies: A(x) = 1 + x*A(x)^2 + x^2/A(x).
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0
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1, 1, 3, 6, 16, 52, 169, 556, 1902, 6667, 23665, 85062, 309444, 1136400, 4205944, 15675690, 58785582, 221644054, 839692735, 3194875093, 12203132521, 46774765206, 179861164503, 693632630011, 2682150035748, 10396942423617, 40393878209664, 157267923352012, 613499893576207
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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: x^2 = A(x)^2 - A(x) - x*A(x)^3.
Recurrence: (n-1)*n*(n+1)*(12*n^2 - 59*n + 54)*a(n) = (n-1)*n*(48*n^3 - 260*n^2 + 331*n - 9)*a(n-1) - (n-1)*(48*n^4 - 332*n^3 + 541*n^2 + 375*n - 648)*a(n-2) + 6*(36*n^5 - 339*n^4 + 1042*n^3 - 1080*n^2 + 59*n + 90)*a(n-3) + 3*(n-33)*n*(3*n-13)*a(n-4) + 9*(n-33)*(n-1)*(3*n - 16)*a(n-5) + 3*n*(3*n-19)*(3*n-14)*(12*n^2 - 35*n + 7)*a(n-6). - Vaclav Kotesovec, Dec 22 2013
a(n) ~ c*d^n/n^(3/2), where d = 4.114104643786649233... is the root of the equation -27 + 27*d - 27*d^2 + 9*d^3 - 5*d^4 + d^5 = 0, and c = 0.595800913094676544457208447889... - Vaclav Kotesovec, Dec 22 2013
Constant c = 1/(2*sqrt(Pi*r)), where r = 0.2241753244886916286... is the root of the equation -1458 + 3645*r + 12420*r^2 + 2448*r^3 - 4610*r^4 + 1383*r^5 = 0. - Vaclav Kotesovec, Jan 04 2014
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 16*x^4 + 52*x^5 + 169*x^6 +...
Related expansions begin:
A(x)^2 = 1 + 2*x + 7*x^2 + 18*x^3 + 53*x^4 + 172*x^5 + 574*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 37*x^3 + 120*x^4 + 405*x^5 + 1398*x^6 +...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^2+x^2/(A+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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