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A209199
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G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*A(x)/A(-x).
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3
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1, 1, 1, 3, 2, 5, 10, 21, 30, 76, 114, 257, 448, 1052, 1706, 4093, 6928, 16284, 28266, 67580, 116288, 278582, 488152, 1168105, 2060388, 4959066, 8772450, 21133812, 37675236, 90901086, 162659624, 393382077, 706479172, 1710430178, 3084264618, 7477512244, 13522121028
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f. satisfies: 1 - x^4 - y + x^2*y + x^3*y^2 - (x*y^2)/(x^2 - y - x*y^2) = 0, where y = A(x). - Vaclav Kotesovec, Mar 13 2014
a(n) ~ (C(r,s1) - (-1)^n*C(-r,s2)) / (sqrt(Pi) * n^(3/2) * r^n), where {r1 = r = 0.45889975689289..., s1 = 3.7914195980097...} and {r2 = -r, s2 = 0.3725313335801...} are roots of the system of equations r^2*(1 + 2*r*s) = 1 + (2*r*s)/(r^2 - s - r*s^2) + (r*s^2*(1 + 2*r*s))/(-r^2 + s + r*s^2)^2, 1 + r^2*s + r^3*s^2 = r^4 + s + (r*s^2)/(r^2 - s - r*s^2), and C(r,s) = sqrt((r*s^2 - r^2 + s)*(4*r^7 - 11*r^6*s^2 - s^3 - 2*r*s^3 - 3*r^4*s^3*(s^3-6) + 10*r^5*s*(s^3-1) - 8*r^3*s^2*(s^3-1) - r^2*s^2*(7*s^2+1)) / (4*r*(r^7 - 3*r^6*s^2 + s^3 - r*s^3 - r^4*s^3*(s^3-6) + 3*r^5*s*(s^3-1) - 3*r^2*s^2*(s^2+1) + r^3*(3*s^2 - 3*s^5 - 1)))), C(r,s1) = 4.083478805997458527..., C(-r,s2) = 0.26836221180354127... - Vaclav Kotesovec, Mar 13 2014
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 10*x^6 + 21*x^7 +...
Related series:
A(x)*A(-x) = 1 + x^2 - x^4 + 5*x^6 + 12*x^8 + 25*x^10 + 164*x^12 +...
A(x)/A(-x) = 1 + 2*x + 2*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 30*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*subst(A, x, -x)+x^2*A/subst(A, x, -x+x*O(x^n))); polcoeff(A, n)}
for(n=0, 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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