A391525 - OEIS

%I #9 Dec 19 2025 08:36:54

%S 1,1,3,13,68,393,2406,15298,99993,667759,4536791,31261365,217958132,

%T 1534798230,10899808305,77978987997,561466298011,4065607240431,

%U 29587409055624,216289975526173,1587512128199805,11694469985169400,86433411486420057,640758350617812090

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

%H Paul D. Hanna, <a href="/A391525/b391525.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

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

%F (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).

%F (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).

%F (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.

%e 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 + ...

%e 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) * ...

%e RELATED SERIES.

%e The self-convolution cube of this sequence yields A391524:

%e 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 + ...

%o (PARI) {a(n) = my(A, N = ceil(log(n+1)/log(3)));

%o A = ((1/x)*serreverse( x / prod(k=0,N, (1 + x^(3^k) +x*O(x^n))^3 )))^(1/3);

%o polcoef(A,n)}

%o for(n=0,30, print1(a(n),", "))

%Y Cf. A391522, A391524.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 18 2025