A369532 G.f. A(x) satisfies: A( 3*A(x)^3 - 54*A(x)^4 ) = 3*x^3.

1, 6, 108, 2514, 66456, 1887624, 56279088, 1737120600, 55031743440, 1779094513494, 58457105824392, 1946499103128888, 65537993150694072, 2227498169071638960, 76321564765011374400, 2633410931198196730680, 91422656110479694644576, 3191108301755033616904896, 111923700012443943176623740

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

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

(1) A( 3*A(x)^3 - 54*A(x)^4 ) = 3*x^3.

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

Example

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 + ...
RELATED SERIES.
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 - ...
where A( 3*A(x)^3 - 54*A(x)^4 ) = 3*x^3.
Let B(x) satisfy A(B(x)) = B(A(x)) = x, where
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 - ...
then B(x)^3 = A(3*x^3 - 54*x^4)/3 which begins
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 + ...

Prog

(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]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

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

Sequence in context: A295791 A361728 A055532 * A288148 A010563 A114310

Adjacent sequences: A369529 A369530 A369531 * A369533 A369534 A369535

Keyword

nonn

Author

Paul D. Hanna, Jan 25 2024

Status

approved