A391524 - OEIS

%I #16 Dec 27 2025 18:05:11

%S 1,3,12,58,318,1887,11772,75969,502554,3389059,23211411,161016948,

%T 1129004057,7988722083,56972470875,409085637498,2955030274440,

%U 21458933307618,156567620461329,1147179838787850,8437546788462108,62273149495359127,461053250613575613,3423339068045408289

%N G.f. A(x) satisfies A(x)^3 = A( x*A(x)^2*(1 + A(x))^3 ).

%C Compare the definition of the g.f. A(x) to the following identities.

%C (1) C(x)^2 = C( x*C(x)*(1 + C(x)) ) where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).

%C (2) F(x)^2 = F( x*F(x)*(1 + F(x))^2 ) where F(x) = x*D(x)^2 and D(x) = 1 + x*D(x)^3 is the g.f. of A001764.

%C (3) G(x)^2 = G( x*G(x)*(1 + G(x))^3 ) where G(x) = x*B(x)^3 and B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

%H Paul D. Hanna, <a href="/A391524/b391524.txt">Table of n, a(n) for n = 1..1000</a>

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

%F (1) A(x) = x * Product_{k>=0} (1 + A(x)^(3^k))^3. Cf. A374628.

%F (2) A(R(x)) = x where R(x) = x / Product_{k>=0} (1 + x^(3^k))^3. Cf. A374627.

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

%F (4) A(x)^9 = A( x*A(x)^8*(1 + A(x))^3*(1 + A(x)^3)^3 ).

%F (5) A(x)^27 = A( x*A(x)^26*(1 + A(x))^3*(1 + A(x)^3)^3*(1 + A(x)^9)^3 ).

%F (6) A(x)^(3^n) = A( x*A(x)^(3^n-1) * Product_{k=0..n-1} (1 + A(x)^(3^k))^3 ).

%F (7) A(x) = x*G(x)^3 where G(x) is the g.f. of A391525.

%F (8) A(x/F(x)) = x*F(x)^2 where F(x) = Product_{k>=0} (1 + (x*F(x)^2)^(3^k)) is the g.f. of A391522.

%F The radius of convergence r of g.f. A(x) and A(r) satisfy 1/3 = Sum_{n>=0} 3^n * A(r)^(3^n) / (1 + A(r)^(3^n)) and r = A(r) / Product_{n>=0} (1 + A(r)^(3^n))^3, where r = 0.126300157685163313827088358997785406302269892311108... and A(r) = 0.316811043952325967889313260937593193863576624981955...

%e G.f.: A(x) = x + 3*x^2 + 12*x^3 + 58*x^4 + 318*x^5 + 1887*x^6 + 11772*x^7 + 75969*x^8 + 502554*x^9 + 3389059*x^10 + ...

%e where A(x)^3 = A( x*A(x)^2*(1 + A(x))^3 ).

%e RELATED SERIES.

%e A(x)^2 = x^2 + 6*x^3 + 33*x^4 + 188*x^5 + 1128*x^6 + 7074*x^7 + 45862*x^8 + 304746*x^9 + 2063466*x^10 + ...

%e A(x)^3 = x^3 + 9*x^4 + 63*x^5 + 417*x^6 + 2754*x^7 + 18423*x^8 + 125112*x^9 + 861300*x^10 + ...

%e This sequence equals the self-convolution cube of A391525 (with offset):

%e (A(x)/x)^(1/3) = 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 + ... + A391525(n)*x^n + ...

%e Also, A(x/F(x)) = x*F(x)^2 where F(x) is the g.f. of A391522:

%e F(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 97*x^5 + 424*x^6 + 1899*x^7 + 8706*x^8 + 40767*x^9 + ... + A391522(n)*x^n + ...

%e and x*F(x)^2 is the g.f. of A391523:

%e x*F(x)^2 = x + 2*x^2 + 5*x^3 + 16*x^4 + 62*x^5 + 264*x^6 + 1170*x^7 + 5310*x^8 + 24599*x^9 + ... + A391523(n)*x^n + ...

%e SPECIFIC VALUES.

%e A(t) = 3/10 at t = 0.126053300834102417758793965089562956754423636918797... where 27/1000 = A(t*19773/100000).

%e A(t) = 1/4 at t = 0.122181330916448478054742245596906752943447511136324... where 1/64 = A(t*125/1024).

%e A(t) = 1/5 at t = 0.113006648268821757608651241461442745445516777004552... where 1/125 = A(t*216/3125).

%e A(t) = 1/6 at t = 0.103511905493866228601748065626664753503783006774793... where 1/216 = A(t*343/7776).

%e A(1/8) = 0.278682168047767664508673941445264399285570529722596...

%e A(1/9) = 0.192464012476338962938895997763792660739217471968762...

%e A(1/10) = 0.156440202717551708260094425960992375254339916967554...

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

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

%o polcoef(A,n)}

%o for(n=1,30, print1(a(n),", "))

%Y Cf. A391522, A391523, A391525, A374627, A374628.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 18 2025