A273958 G.f. A(x) satisfies: x*A(x) + x^2*A(x)^2 = C(x)^2, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

1, 2, 4, 10, 30, 96, 313, 1038, 3512, 12100, 42286, 149440, 533076, 1917000, 6942660, 25300034, 92701746, 341319648, 1262176462, 4685724500, 17457027376, 65247128392, 244584468174, 919319871360, 3464027857900, 13082491517336, 49513139090512, 187761168576904, 713325243149880, 2714644395230080, 10347465825694361, 39500936075084958, 151005816167189208, 578038682649831252, 2215461387843704450, 8501300598574467776, 32658238309503587964

Offset: 1

Author

Paul D. Hanna, Jun 10 2016

Keywords

• nonn

Comments

It appears that a(n) = 1 (mod 2) iff n = 2*4^k - 1 for k>=0.

Radius of convergence is 1/4 with A(1/4) = 2*(sqrt(2) - 1).

Examples

			G.f.: A(x) = x + 2*x^2 + 4*x^3 + 10*x^4 + 30*x^5 + 96*x^6 + 313*x^7 + 1038*x^8 + 3512*x^9 + 12100*x^10 + 42286*x^11 + 149440*x^12 + +...
such that x*A(x) + x^2*A(x)^2 = C(x)^2, where
A(x)^2 = x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 116*x^6 + 392*x^7 + 1350*x^8 + 4696*x^9 + 16500*x^10 + 58572*x^11 + 209824*x^12 +...+ A055395(n+2)*x^n +...
C(x)^2 = x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 +...+ A000108(n)*x^n +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..520

Crossrefs

Cf. A055395.

Programs

PARI

/* From A(x) = (sqrt(1 + 4*C(x)^2) - 1)/(2*x) */
{a(n) = my(A=x,C=(1 - sqrt(1 - 4*x +x*O(x^n)))/2); A = (sqrt(1 + 4*C^2) - 1)/(2*x); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

PARI

/* From A(x) = (1/x)*Series_Reversion( sqrt(x^2 + x^4) - (x^2 + x^4) )^2 */
{a(n) = my(A=x); A = ( serreverse( sqrt(x^2 + x^4 +x^3*O(x^n)) - (x^2 + x^4) ) )^2/x; polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = (sqrt(1 + 4*C(x)^2) - 1)/(2*x), where C(x) = (1 - sqrt(1 - 4*x))/2.

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

a(n) ~ 2^(2*n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 23 2016