OFFSET
0,3
COMMENTS
See A383553 for more formulas involving the g.f. of this sequence.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. C(x) = Sum_{n>=0} a(n)*x^n along with related series A(x) (A383553) and B(x) (A383554) satisfy the following formulas.
(1) C(x) = (1 - x/C(x)) * C(x/C(x))^2.
(2.a) C(x)^2 = C( x*(1-x)*C(x)^2 ) / (1-x).
(2.b) C(x)^2 = Series_Reversion(x/C(x)) / (x - x^2).
(3) [x^n] x/C(x)^2 = (1/n) * [x^n] x/A(x)^(2*n) for n >= 1.
(4) C(x)^2 = B(x)/(1-x).
(5) C(x) = A( x/C(x)^2 ).
(6) C(x) = B(x/C(x)).
(7) A(x) = C(x*A(x)^2) is the g.f. of A383553.
(8) B(x) = C(x*B(x)) is the g.f. of A383554.
(9) C(x) = B( x*sqrt((1-x)/B(x)) ).
(10.a) C(x*A(x)) = A( x*(1 - x*A(x)) ).
(10.b) C(x*A(x))^2 = A(x) / (1 - x*A(x)).
a(n) ~ c * n! * n^(2*log(2)) / log(2)^n, where c = 0.0907361443240267910478474... - Vaclav Kotesovec, Jun 09 2025
EXAMPLE
G.f.: C(x) = 1 + x + 2*x^2 + 8*x^3 + 53*x^4 + 474*x^5 + 5160*x^6 + 65044*x^7 + 923050*x^8 + 14485824*x^9 + 248342433*x^10 + ...
where C(x) = (1 - x/C(x)) * C(x/C(x))^2.
RELATED SERIES.
A(x) = 1 + x + 4*x^2 + 25*x^3 + 203*x^4 + 1986*x^5 + 22492*x^6 + 287779*x^7 + 4092708*x^8 + ... + A383553(n)*x^n ...
where A(x) = C(x*A(x)^2),
also, A(x) = (1 - x*A(x)) * A( x*(1 - x*A(x)) )^2.
B(x) = 1 + x + 3*x^2 + 15*x^3 + 106*x^4 + 960*x^5 + 10458*x^6 + 131608*x^7 + 1864069*x^8 + ... + A383554(n)*x^n ...
where B(x) = C(x*B(x)),
also, B(x)^2 = B(x*B(x)) / (1 - x*B(x)).
PROG
(PARI) {a(n) = my(V=[1]); for(i=0, n, V = concat(V, 0); B = Ser(V);
V[#V] = 2*polcoef( sqrt(serreverse(x/C)/(x-x^2)) - C, #V-2); ); polcoef(C, n)}
for(n=0, 21, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 03 2025
STATUS
approved