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. B(x) = Sum_{n>=0} a(n)*x^n along with related series A(x) (A383553) and C(x) (A383555) satisfy the following formulas.
(1) B(x)^2 = B(x*B(x)) / (1 - x*B(x)).
(2.a) B(x) = (1-x) * B( x*sqrt((1-x)/B(x)) )^2.
(2.b) B(x) = (1-x) * ( x/Series_Reversion(x*B(x)) )^2.
(3) [x^n] x/B(x) = (1/n) * [x^n] x/A(x)^n for n >= 1.
(4) B(x) = (1-x) * C(x)^2.
(5) B(x) = A(x/B(x)).
(6) B(x) = C(x*B(x)).
(7) A(x) = B(x*A(x)) is the g.f. of A383553.
(8) C(x) = B(x/C(x)) is the g.f. of A383555.
(9) B(x) = C( x*(1-x)*C(x)^2 ).
(10.a) B(x*A(x)^2) = A( x/(1 - x*A(x)^2) ).
(10.b) B(x*A(x)^2) = A(x)^2 * (1 - x*A(x)^2).
a(n) ~ c * n! * n^(2*log(2)) / log(2)^n, where c = 0.181472288648053582095694... = 2 * c_A383555. - Vaclav Kotesovec, Jun 09 2025
EXAMPLE
G.f.: 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 + 29203507*x^9 + 499950326*x^10 + ...
where B(x)^2 = B(x*B(x)) / (1 - x*B(x)).
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) = B(x*A(x)),
also, A(x) = (1 - x*A(x)) * A( x*(1 - x*A(x)) )^2.
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 + ... + A383555(n)*x^n + ...
where C(x) = B(x/C(x)),
also, C(x) = (1 - x/C(x)) * C(x/C(x))^2.
PROG
(PARI) {a(n) = my(V=[1]); for(i=0, n, V = concat(V, 0); B = Ser(V);
V[#V] = polcoef( B - (1-x)*(x/serreverse(x*B))^2, #V-1); ); H=B; polcoef(B, n)}
for(n=0, 21, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 03 2025
STATUS
approved