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%I #9 Oct 26 2016 17:18:42
%S 1,1,10,141,2422,47562,1031764,24214405,606444990,16055089470,
%T 446238074892,12955112773554,391332183548956,12261884937532340,
%U 397576302315045800,13313017677172350965,459635990935574444942,16339309997761322057206,597340515437542895494748,22435278085988347895795526,864900964565994975048855444,34195693888939483596581262668,1385553440866978431053220575128
%N G.f. satisfies: A(x - 4*A(x)^2) = x - 3*A(x)^2.
%H Paul D. Hanna, <a href="/A277310/b277310.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) also satisfies:
%F (1) A(x) = x + A( 4*A(x) - 3*x )^2.
%F (2) A(x) = 3*x/4 + 1/4 * Series_Reversion(x - 4*A(x)^2).
%F (3) R(x) = 4*x/3 - 1/3 * Series_Reversion(x - 3*A(x)^2), where R(A(x)) = x.
%F (4) R( sqrt( x - R(x) ) ) = 4*x - 3*R(x), where R(A(x)) = x.
%F (5) A(x) = x + Sum_{n>=1} 4^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
%F a(n) = Sum_{k=0..n-1} A277295(n,k) * 4^k.
%e G.f.: A(x) = x + x^2 + 10*x^3 + 141*x^4 + 2422*x^5 + 47562*x^6 + 1031764*x^7 + 24214405*x^8 + 606444990*x^9 + 16055089470*x^10 +...
%e such that A(x - 4*A(x)^2) = x - 3*A(x)^2.
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^3 + 21*x^4 + 302*x^5 + 5226*x^6 + 102788*x^7 + 2226973*x^8 + 52126582*x^9 + 1301232638*x^10 + 34328704796*x^11 + 950803699394*x^12 + 27510261070028*x^13 + 828332416917876*x^14 + 25876801064095496*x^15 + 836682915170627501*x^16 +...
%e A(x - 4*A(x)^2) = x - 3*x^2 - 6*x^3 - 63*x^4 - 906*x^5 - 15678*x^6 - 308364*x^7 - 6680919*x^8 - 156379746*x^9 - 3903697914*x^10 +...
%e which equals x - 3*A(x)^2.
%e Series_Reversion(x - 4*A(x)^2) = x + 4*x^2 + 40*x^3 + 564*x^4 + 9688*x^5 + 190248*x^6 + 4127056*x^7 + 96857620*x^8 + 2425779960*x^9 + 64220357880*x^10 +...
%e which equals -3*x + 4*A(x).
%e A( 4*A(x) - 3*x ) = x + 5*x^2 + 58*x^3 + 921*x^4 + 17494*x^5 + 374994*x^6 + 8793460*x^7 + 221393569*x^8 + 5912166718*x^9 + 166058455158*x^10 + 4876311925036*x^11 + 149037482367530*x^12 + 4724877954111836*x^13 + 154959634972646340*x^14 + 5246331138228520168*x^15 +...
%e which equals sqrt( A(x) - x ).
%o (PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-4*F^2) + 3*F^2, #A) ); A[n]}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
%Y Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Oct 12 2016
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