A363560 Expansion of g.f. A(x) satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).

1, 1, 3, 18, 126, 966, 7863, 66696, 583111, 5217513, 47547405, 439777242, 4117802109, 38956162023, 371795456373, 3575401032544, 34611064585803, 336998629754631, 3298200003722997, 32428037256038775, 320151289224740949, 3172536384239678856, 31544584654878015766

Offset: 0

Author

Paul D. Hanna, Aug 12 2023

Keywords

• nonn

Comments

Compare to: G(x)^3 = 1 + x*(G(x) + G(x)^2 + G(x)^3) holds when G(x) = 1/(1-x).

Conjecture: a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 126*x^4 + 966*x^5 + 7863*x^6 + 66696*x^7 + 583111*x^8 + 5217513*x^9 + 47547405*x^10 + ...
such that
A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).
Also,
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
RELATED TABLE.
The table of coefficients in A(x)^n begins:
n=1: [1, 1,  3,  18,  126,   966,   7863,   66696, ...];
n=2: [1, 2,  7,  42,  297,  2292,  18738,  159450, ...];
n=3: [1, 3, 12,  73,  522,  4059,  33354,  284886, ...];
n=4: [1, 4, 18, 112,  811,  6360,  52566,  450888, ...];
n=5: [1, 5, 25, 160, 1175,  9301,  77370,  666780, ...];
n=6: [1, 6, 33, 218, 1626, 13002, 108919,  943524, ...];
n=7: [1, 7, 42, 287, 2177, 17598, 148540, 1293937, ...];
n=8: [1, 8, 52, 368, 2842, 23240, 197752, 1732928, ...];
n=9: [1, 9, 63, 462, 3636, 30096, 258285, 2277756, ...];
...
from which one can verify the formulas involving powers of A(x).
RELATED SERIES.
Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^5) )
where
G(x) = 1 + x + 2*x^2 + 11*x^3 + 61*x^4 + 380*x^5 + 2502*x^6 + 17163*x^7 + 121312*x^8 + 877370*x^9 + 6461765*x^10 + ...
then
A(x) = G(x*A(x)),
and so
A(x) = (1/x) * Series_Reversion( x/G(x) );
thus,
x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^5) )
which is equivalent to
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
TERMS MODULO 3.
It appears that a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).
The residues of a(7*k + 1) modulo 3, for k >= 0, begin
a(7*k + 1) (mod 3) = [1, 1, 1, 1, 0, 2, 1, 0, 0, 1, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A300048, A161634.

Programs

PARI

{a(n) = my(A=1+x); for(i=1, n, A = (1 + x*(A + A^2 + A^9) +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

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

(1) A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).

(2) A(x)^2 = 1 + x*(A(x) + A(x)^2 - A(x)^3 + A(x)^5 - A(x)^6 + A(x)^8) .

(3) A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).

(4) A(x)^4 = 1 + x*(A(x) + A(x)^2 + A(x)^4 - A(x)^6 + A(x)^7 + A(x)^10).

(5) A(x)^5 = 1 + x*(A(x) + A(x)^2 + A(x)^4 + A(x)^5 - A(x)^6 + A(x)^8 + A(x)^11).

(6) A(x)^6 = 1 + x*(A(x) + A(x)^2 + A(x)^4 + A(x)^5 + A(x)^9 + A(x)^12).

(7) A(x) = (1/x) * Series_Reversion( x/(1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^5) ) ) ).