login
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).
2

%I #18 Dec 04 2022 08:34:20

%S 1,1,4,15,65,291,1356,6474,31555,156315,784924,3986534,20444676,

%T 105728100,550735400,2886924190,15217019595,80600822575,428766983300,

%U 2289637381800,12268642450420,65941128441080,355396218177760,1920215555772550,10398415258863275

%N 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).

%C Self convolution equals A357547.

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

%C Related identities:

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

%C (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).

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

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

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

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

%F (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).

%e 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 + ...

%e such that

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

%e where

%e 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) + ...

%o (PARI) {a(n) = my(A=x); for(i=1, #binary(n+1),

%o 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)) )

%o ); polcoeff(H=A, n)}

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A357547, A357786.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Dec 03 2022