A370540 Expansion of g.f. A(x) satisfying A(x)^2 = A(x^2) * (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

1, 1, 4, 12, 45, 157, 584, 2155, 8110, 30587, 116326, 443984, 1702272, 6546563, 25252094, 97638658, 378351696, 1468876958, 5712276601, 22247635905, 86765271643, 338795469496, 1324374411164, 5182303804184, 20297243177269, 79564763550396, 312137086267106, 1225421059470049

Offset: 0

Author

Paul D. Hanna, Mar 12 2024

Keywords

• nonn

Examples

			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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..600

Crossrefs

Cf. A370539, A356781, A000108.

Programs

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),", "))

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