A357785 a(n) = coefficient of x^n, n >= 1, in A(x) such that: A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ) * sqrt(1 - 4*x - 4*x^2).

1, 1, 4, 15, 65, 291, 1356, 6474, 31555, 156315, 784924, 3986534, 20444676, 105728100, 550735400, 2886924190, 15217019595, 80600822575, 428766983300, 2289637381800, 12268642450420, 65941128441080, 355396218177760, 1920215555772550, 10398415258863275

Offset: 1

Author

Paul D. Hanna, Dec 03 2022

Keywords

• nonn

Comments

Self convolution equals A357547.

Radius of convergence is r = (sqrt(41) - 5)/8, where r = r^2/(1 - 4*r - 4*r^2), with A(r) = sqrt(r).

Related identities:

(1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).

(2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 15*x^4 + 65*x^5 + 291*x^6 + 1356*x^7 + 6474*x^8 + 31555*x^9 + 156315*x^10 + 784924*x^11 + 3986534*x^12 + ...
such that
A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ) * sqrt(1 - 4*x - 4*x^2)
where
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 38*x^5 + 176*x^6 + 832*x^7 + 4039*x^8 + 19938*x^9 + 99861*x^10 + ... + A357547(n)*x^(n+1) + ...
		

Crossrefs

Cf. A357547, A357786.

Programs

PARI

{a(n) = my(A=x); for(i=1, #binary(n+1),
A = sqrt( subst(A, x, x^2/(1 - 4*x - 4*x^2 +x*O(x^n)) )*sqrt(1 - 4*x - 4*x^2 +x*O(x^n)) )
); polcoeff(H=A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

(1) A(x) = -A( -x/(1 - 4*x) ) * sqrt(1 - 4*x).

(2) A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ) * sqrt(1 - 4*x - 4*x^2).

(3) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 8*x^2) ) * sqrt(1 - 8*x^2) / (1 + 2*x).

(4) A( x/(1 + 2*x + 6*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 6^2*x^4) ) * sqrt(1 + 2^2*x^2 + 6^2*x^4) / (1 + 2*x + 6*x^2).