login
G.f. A(x) satisfies: A( 3*A(x)^3 - 54*A(x)^4 ) = 3*x^3.
5

%I #7 Jan 26 2024 08:35:36

%S 1,6,108,2514,66456,1887624,56279088,1737120600,55031743440,

%T 1779094513494,58457105824392,1946499103128888,65537993150694072,

%U 2227498169071638960,76321564765011374400,2633410931198196730680,91422656110479694644576,3191108301755033616904896,111923700012443943176623740

%N G.f. A(x) satisfies: A( 3*A(x)^3 - 54*A(x)^4 ) = 3*x^3.

%H Paul D. Hanna, <a href="/A369532/b369532.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 - 54*A(x)^4 ) = 3*x^3.

%F (2) A( ( A(3*x^3 - 54*x^4)/3 )^(1/3) ) = x.

%e G.f.: A(x) = x + 6*x^2 + 108*x^3 + 2514*x^4 + 66456*x^5 + 1887624*x^6 + 56279088*x^7 + 1737120600*x^8 + 55031743440*x^9 + 1779094513494*x^10 + ...

%e RELATED SERIES.

%e 3*A(x)^3 - 54*A(x)^4 = 3*x^3 - 54*x^6 - 972*x^9 - 28674*x^12 - 1084752*x^15 - 41255568*x^18 - 1741551840*x^21 - 76317761952*x^24 - ...

%e where A( 3*A(x)^3 - 54*A(x)^4 ) = 3*x^3.

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

%e B(x) = x - 6*x^2 - 36*x^3 - 354*x^4 - 4464*x^5 - 56592*x^6 - 796320*x^7 - 11632032*x^8 - 174223872*x^9 - 2674468710*x^10 - ...

%e then B(x)^3 = A(3*x^3 - 54*x^4)/3 which begins

%e B(x)^3 = x^3 - 18*x^4 + 18*x^6 - 648*x^7 + 5832*x^8 + 972*x^9 - 52488*x^10 + 944784*x^11 - 5600826*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 - 54*A^4 ), #V+2)/9); V[n]}

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

%Y Cf. A291315, A369533, A369531, A369534, A369535.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 25 2024