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A271935
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G.f. A(x) satisfies: A(x) = A( x^2 + 8*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.
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4
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1, 4, 26, 200, 1691, 15204, 142710, 1382568, 13721765, 138802136, 1425785270, 14832383488, 155947271878, 1654494195340, 17690004381000, 190426309700616, 2062071992480208, 22447191471665160, 245501068961175090, 2696300196714320520, 29725402250477117175, 328835072363241763920, 3649104297070240016280, 40610292053766375861000, 453133171709490497936328, 5068340076350484467559468
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OFFSET
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1,2
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COMMENTS
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Compare the g.f. to the following related identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).
(2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).
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LINKS
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FORMULA
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G.f. A(x) satisfies: A( x*G(x^2) - 4*x^2 ) = x, where G(x)^2 = G(x^2) + 12*x, and G(x) is the g.f. of A264413.
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EXAMPLE
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G..f.: A(x) = x + 4*x^2 + 26*x^3 + 200*x^4 + 1691*x^5 + 15204*x^6 + 142710*x^7 + 1382568*x^8 + 13721765*x^9 + 138802136*x^10 + 1425785270*x^11 + 14832383488*x^12 +...
where A(x)^2 = A( x^2 + 8*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 8*x^3 + 68*x^4 + 608*x^5 + 5658*x^6 + 54336*x^7 + 534984*x^8 + 5373824*x^9 + 54866075*x^10 + 567775856*x^11 + 5942353444*x^12 +...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 4*x^2 + 6*x^3 - 15*x^5 + 90*x^7 - 660*x^9 + 5310*x^11 - 45765*x^13 + 413640*x^15 - 3864345*x^17 + 37014120*x^19 +...+ A264413(n)*x^(2*n+1) +...
such that B(x) = x*G(x^2) - 4*x^2 where G(x)^2 = G(x^2) + 12*x, and G(x) is the g.f. of A264413.
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PROG
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(PARI) {a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, x^2 + 8*X*A^2)^(1/2) ); polcoeff(A, n)}
for(n=1, 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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