A277303 - OEIS

%I #9 Oct 25 2016 21:15:43

%S 1,5,90,2425,80630,3065810,128271540,5774538945,275743894750,

%T 13832116773110,723891526915820,39323723086794730,2208811824884144540,

%U 127904686371063157700,7617441454740093233000,465691699545009287055825,29179499379365501297165550,1871486497257264286902367950,122731222232573572625823907900,8222122259910817121846641763950,562251437460415648354364719018900

%N G.f. satisfies: A(x - 4*A(x)^2) = x + A(x)^2.

%H Paul D. Hanna, <a href="/A277303/b277303.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) also satisfies:

%F (1) A(x) = x + 5 * A( x/5 + 4*A(x)/5 )^2.

%F (2) A(x) = -x/4 + 5/4 * Series_Reversion(x - 4*A(x)^2).

%F (3) R(x) = -4*x + 5 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.

%F (4) R( sqrt( x/5 - R(x)/5 ) ) = 4*x/5 + R(x)/5, where R(A(x)) = x.

%F a(n) = Sum_{k=0..n-1} A277295(n,k) * 4^k * 5^(n-k-1).

%e G.f.: A(x) = x + 5*x^2 + 90*x^3 + 2425*x^4 + 80630*x^5 + 3065810*x^6 + 128271540*x^7 + 5774538945*x^8 + 275743894750*x^9 + 13832116773110*x^10 +...

%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) - 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, A277304, A277305, A277306, A277307, A277308, A277309.

%Y Cf. A276364.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 09 2016