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..1500
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))^2. Cf. A120880.
(2) A(R(x)) = x where R(x) = x / Product_{k>=0} (1 + x^(3^k))^2. Cf. A309677.
(3) A(x) = x * Sum_{n>=0} 2^A062756(n) * A(x)^n, where A062756(n) is the number of 1's in the ternary expansion of n.
(4) A(x)^3 = A( x*A(x)^2*(1 + A(x))^2 ).
(5) A(x)^9 = A( x*A(x)^8*(1 + A(x))^2*(1 + A(x)^3)^2 ).
(6) A(x)^27 = A( x*A(x)^26*(1 + A(x))^2*(1 + A(x)^3)^2*(1 + A(x)^9)^2 ).
(7) A(x)^(3^n) = A( x*A(x)^(3^n-1) * Product_{k=0..n-1} (1 + A(x)^(3^k))^2 ) for n > 0.
The radius of convergence r of g.f. A(x) and A(r) satisfy 1/2 = 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))^2, where r = 0.1804102993953842577608109614601127784696416600317359... and A(r) = 0.416719679127135468599692361809894039884295747411470...
EXAMPLE
G.f.: A(x) = 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 + 116092*x^10 + ...
where A(x)^3 = A( x*A(x)^2*(1 + A(x))^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 14*x^4 + 52*x^5 + 213*x^6 + 936*x^7 + 4272*x^8 + 19924*x^9 + 94430*x^10 + ...
A(x)^3 = x^3 + 6*x^4 + 27*x^5 + 116*x^6 + 513*x^7 + 2358*x^8 + 11135*x^9 + 53466*x^10 + ...
This sequence equals the self-convolution of A391522 (with offset):
(A(x)/x)^(1/2) = 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 + ...
SPECIFIC VALUES.
A(t) = 2/5 at t = 0.18017436432346732248567020355142360233078350225996... where 8/125 = A(t*196/625).
A(t) = 1/3 at t = 0.17432858695082469181616483229105669565541944345209... where 1/27 = A(t*16/81).
A(t) = 1/4 at t = 0.15511360947422722158928001634253620061940194882675... where 1/64 = A(t*25/256).
A(t) = 1/5 at t = 0.13669291173270177195677173147159425596375285303138... where 1/125 = A(t*36/625).
A(t) = 1/6 at t = 0.12132299386374854802895920406601523316655862672518... where 1/216 = A(t*49/1296).
A(1/6) = 0.29261216811670615422839774118614212104203206030845...
A(1/7) = 0.21517440625215536825526473529625335449966184809762...
A(1/8) = 0.17415557848292409758231994304176593189932373590367...
A(1/9) = 0.14715038310622855597708357001689002569103128389917...
A(1/10) = 0.12770122705800330694756977616690931749345774351589...
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))^2 ));
polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 14 2025
STATUS
approved