OFFSET
1,2
COMMENTS
Compare the definition of the g.f. A(x) to the following identities.
(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).
(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.
(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.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1000
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = x * Product_{k>=0} (1 + A(x)^(3^k))^3. Cf. A374628.
(2) A(R(x)) = x where R(x) = x / Product_{k>=0} (1 + x^(3^k))^3. Cf. A374627.
(3) A(x)^3 = A( x*A(x)^2*(1 + A(x))^3 ).
(4) A(x)^9 = A( x*A(x)^8*(1 + A(x))^3*(1 + A(x)^3)^3 ).
(5) A(x)^27 = A( x*A(x)^26*(1 + A(x))^3*(1 + A(x)^3)^3*(1 + A(x)^9)^3 ).
(6) A(x)^(3^n) = A( x*A(x)^(3^n-1) * Product_{k=0..n-1} (1 + A(x)^(3^k))^3 ).
(7) A(x) = x*G(x)^3 where G(x) is the g.f. of A391525.
(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.
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...
EXAMPLE
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 + ...
where A(x)^3 = A( x*A(x)^2*(1 + A(x))^3 ).
RELATED SERIES.
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 + ...
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 + ...
This sequence equals the self-convolution cube of A391525 (with offset):
(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 + ...
Also, A(x/F(x)) = x*F(x)^2 where F(x) is the g.f. of A391522:
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 + ...
and x*F(x)^2 is the g.f. of A391523:
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 + ...
SPECIFIC VALUES.
A(t) = 3/10 at t = 0.126053300834102417758793965089562956754423636918797... where 27/1000 = A(t*19773/100000).
A(t) = 1/4 at t = 0.122181330916448478054742245596906752943447511136324... where 1/64 = A(t*125/1024).
A(t) = 1/5 at t = 0.113006648268821757608651241461442745445516777004552... where 1/125 = A(t*216/3125).
A(t) = 1/6 at t = 0.103511905493866228601748065626664753503783006774793... where 1/216 = A(t*343/7776).
A(1/8) = 0.278682168047767664508673941445264399285570529722596...
A(1/9) = 0.192464012476338962938895997763792660739217471968762...
A(1/10) = 0.156440202717551708260094425960992375254339916967554...
PROG
(PARI) {a(n) = my(A, N = ceil(log(n+1)/log(3)));
A = serreverse( x / prod(k=0, N, (1 + x^(3^k) +x*O(x^n))^3 ));
polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 18 2025
STATUS
approved