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G.f. A(x) satisfies: A( 3*A(x)^3 - 81*A(x)^4 ) = 3*x^3.
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%I #7 Jan 26 2024 08:35:27

%S 1,9,243,8496,336636,14342346,641438028,29699299620,1411372664280,

%T 68444935282701,3373606703086362,168510442754976957,

%U 8511020689545160092,433932382981014590340,22303262300042449394280,1154400083338031391572595,60118518601311326617518846,3147841114874882473908795039

%N G.f. A(x) satisfies: A( 3*A(x)^3 - 81*A(x)^4 ) = 3*x^3.

%H Paul D. Hanna, <a href="/A369533/b369533.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) A( 3*A(x)^3 - 81*A(x)^4 ) = 3*x^3.

%F (2) A( ( A(3*x^3 - 81*x^4)/3 )^(1/3) ) = x.

%e G.f.: A(x) = x + 9*x^2 + 243*x^3 + 8496*x^4 + 336636*x^5 + 14342346*x^6 + 641438028*x^7 + 29699299620*x^8 + 1411372664280*x^9 + ...

%e RELATED SERIES.

%e 3*A(x)^3 - 81*A(x)^4 = 3*x^3 - 81*x^6 - 2187*x^9 - 97686*x^12 - 5393142*x^15 - 314613072*x^18 - 19867036050*x^21 - ...

%e where A( 3*A(x)^3 - 81*A(x)^4 ) = 3*x^3.

%e Let B(x) satisfy A(B(x)) = B(A(x)) = x, where

%e B(x) = x - 9*x^2 - 81*x^3 - 1206*x^4 - 22194*x^5 - 431568*x^6 - 9084150*x^7 - 198701343*x^8 - 4469924007*x^9 - ...

%e then B(x)^3 = A(3*x^3 - 81*x^4)/3 which begins

%e B(x)^3 = x^3 - 27*x^4 + 27*x^6 - 1458*x^7 + 19683*x^8 + 2187*x^9 - 177147*x^10 + 4782969*x^11 - 42817329*x^12 + ...

%o (PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = x*Ser(V); V[#V] = -polcoeff(subst(G=A, x, 3*A^3 - 81*A^4 ), #V+2)/9); V[n]}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A291315, A369532, A369531, A369534, A369535.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 25 2024