A391525 G.f. satisfies A(x) = Product_{n>=0} (1 + (x*A(x)^3)^(3^n)).

1, 1, 3, 13, 68, 393, 2406, 15298, 99993, 667759, 4536791, 31261365, 217958132, 1534798230, 10899808305, 77978987997, 561466298011, 4065607240431, 29587409055624, 216289975526173, 1587512128199805, 11694469985169400, 86433411486420057, 640758350617812090

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = Product_{n>=0} (1 + (x*A(x)^3)^(3^n)).

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

(3) A(x)^3 = (1/x) * Series_Reversion( x / Product_{n>=0} (1 + x^(3^n))^3 ).

(4) A(x) = P * A( x^9*A(x)^24 * P^3 ) where P = (1 + x*A(x)^3)*(1 + x^3*A(x)^9).

(5) A(x) = P * A( x^27*A(x)^78 * P^3 ) where P = (1 + x*A(x)^3)*(1 + x^3*A(x)^9)*(1 + x^9*A(x)^27).

(6) A(x) = P * A( x^(3^n)*A(x)^(3*3^n-3) * P^3 ) where P = Product_{k=0..n-1} (1 + (x*A(x)^3)^(3^k)) for n >= 1.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 393*x^5 + 2406*x^6 + 15298*x^7 + 99993*x^8 + 667759*x^9 + 4536791*x^10 + ...
where A(x) = (1 + x*A(x)^3) * (1 + x^3*A(x)^9) * (1 + x^9*A(x)^27) * (1 + x^27*A(x)^81) * ...
RELATED SERIES.
The self-convolution cube of this sequence yields A391524:
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 318*x^4 + 1887*x^5 + 11772*x^6 + 75969*x^7 + 502554*x^8 + ... + A391524(n+1)*x^n + ...

Prog

(PARI) {a(n) = my(A, N = ceil(log(n+1)/log(3)));
A = ((1/x)*serreverse( x / prod(k=0, N, (1 + x^(3^k) +x*O(x^n))^3 )))^(1/3);
polcoef(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A391522, A391524.

Sequence in context: A060014 A182666 A042659 * A054132 A047149 A200757

Adjacent sequences: A391522 A391523 A391524 * A391526 A391527 A391528

Keyword

nonn

Author

Paul D. Hanna, Dec 18 2025

Status

approved