A369533 G.f. A(x) satisfies: A( 3*A(x)^3 - 81*A(x)^4 ) = 3*x^3.

1, 9, 243, 8496, 336636, 14342346, 641438028, 29699299620, 1411372664280, 68444935282701, 3373606703086362, 168510442754976957, 8511020689545160092, 433932382981014590340, 22303262300042449394280, 1154400083338031391572595, 60118518601311326617518846, 3147841114874882473908795039

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

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

Example

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 + ...
RELATED SERIES.
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 - ...
where A( 3*A(x)^3 - 81*A(x)^4 ) = 3*x^3.
Let B(x) satisfy A(B(x)) = B(A(x)) = x, where
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 - ...
then B(x)^3 = A(3*x^3 - 81*x^4)/3 which begins
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 + ...

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 - 81*A^4 ), #V+2)/9); V[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

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

Sequence in context: A268088 A285511 A013733 * A295228 A217628 A167005

Adjacent sequences: A369530 A369531 A369532 * A369534 A369535 A369536

Keyword

nonn

Author

Paul D. Hanna, Jan 25 2024

Status

approved