OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..600
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n and C(x) = (1 - sqrt(1-4*x))/2 satisfy the following formulas.
(1) A(x)^2 = A(x^2) * F(x) where F(x) = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) is the g.f. of A370539.
(2) G( x*A(x^2)*(1 - x*C(x^2)) ) = x, where G(x) = G( x^2 + 2*x^2*G(x) )^(1/2) is the g.f. of A356781.
a(n) ~ c * 4^n / sqrt(n), where c = 0.3550434768046000612979284344613941075803... - Vaclav Kotesovec, Mar 14 2024
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 12*x^3 + 45*x^4 + 157*x^5 + 584*x^6 + 2155*x^7 + 8110*x^8 + 30587*x^9 + 116326*x^10 + 443984*x^11 + ...
RELATED SERIES.
We may illustrate the formulas using the following related series expansions.
Recall that the Catalan function C(x) = (1 - sqrt(1-4*x))/2 begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ... + A000108(n)*x^n + ...
(1) By definition, A(x) = sqrt( A(x^2) * F(x) ) where
F(x) = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) begins
F(x) = 1 + 2*x + 8*x^2 + 30*x^3 + 118*x^4 + 462*x^5 + 1824*x^6 + 7208*x^7 + 28558*x^8 + ... + A370539(n)*x^n + ...
(2) Also, G(x) = G( x^2 + 2*x^2*G(x) )^(1/2) begins
G(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 14*x^7 + 32*x^8 + 74*x^9 + 172*x^10 + 408*x^11 + ... + A356781(n)*x^n + ...
such that the series reversion of G(x) equals
x*A(x^2)*(1 - x*C(x^2)) = x - x^2 + x^3 - 2*x^4 + 4*x^5 - 7*x^6 + 12*x^7 - 23*x^8 + 45*x^9 - 84*x^10 + 157*x^11 - 302*x^12 + 584*x^13 - 1121*x^14 + ...
PROG
(PARI) {a(n) = my(x = 'x + O('x^(n+4)), C(x) = (1 - sqrt(1 - 4*x))/(2*x), A = 1+x); for(i=1, n, A = sqrt( subst(A, 'x, x^2) * (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) ) ); polcoeff(A, n); }
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 12 2024
STATUS
approved