|
%I #9 Oct 25 2016 21:22:38
%S 1,2,20,298,5492,116124,2710776,68308170,1831522940,51744512380,
%T 1529687560328,47075470016012,1502258036769256,49560341916549320,
%U 1686236991420431760,59054595629732284890,2125432920387784135812,78509698415432235272292,2972996232264052816975752,115303660044380692013332428
%N G.f. satisfies: A(x - 3*A(x)^2) = x - A(x)^2.
%H Paul D. Hanna, <a href="/A277308/b277308.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) also satisfies:
%F (1) A(x) = x + 2 * A( 3*A(x)/2 - x/2 )^2.
%F (2) A(x) = x/3 + 2/3 * Series_Reversion(x - 3*A(x)^2).
%F (3) R(x) = 3*x - 2 * Series_Reversion(x - A(x)^2), where R(A(x)) = x.
%F (4) R( sqrt( x/2 - R(x)/2 ) ) = 3*x/2 - R(x)/2, where R(A(x)) = x.
%F a(n) = Sum_{k=0..n-1} A277295(n,k) * 3^k * 2^(n-k-1).
%e G.f.: A(x) = x + 2*x^2 + 20*x^3 + 298*x^4 + 5492*x^5 + 116124*x^6 + 2710776*x^7 + 68308170*x^8 + 1831522940*x^9 + 51744512380*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-3*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, A277303, A277304, A277305, A277306, A277307, A277309.
%Y Cf. A276364.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Oct 09 2016
|
