A200376 G.f.: 1/sqrt(1-10*x^2 + x^4/(1-8*x^2)) + x/(1-9*x^2).

1, 1, 5, 9, 37, 81, 301, 729, 2549, 6561, 22045, 59049, 193029, 531441, 1703469, 4782969, 15111573, 43046721, 134539837, 387420489, 1200901157, 3486784401, 10739313997, 31381059609, 96172251061, 282429536481, 862142190941, 2541865828329, 7734936371269, 22876792454961, 69439155241581

Offset: 0

Author

Paul D. Hanna, Nov 16 2011

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...
The g.f. of A200375(n) = A000108(n)*A001045(n) begins:
G(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 +...
where A(x) = G(x/A(x)) and G(x) = A(x*G(x)).
		

Crossrefs

Cf. A200375, A098615, A098617.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1-10x^2+x^4/(1-8x^2)]+x/(1-9x^2),{x,0,30}], x] (* Harvey P. Dale, Nov 19 2011 *)

PARI

{a(n)=polcoeff(1/sqrt(1-10*x^2 + x^4/(1-8*x^2 +x*O(x^n))) + x/(1-9*x^2 +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

PARI

{a(n)=local(G=sum(m=0,n,binomial(2*m, m)/(m+1)*polcoeff(1/(1-x-2*x^2+x*O(x^m)), m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G),n)}
for(n=0,30,print1(a(n),", "))

Formula

D-finite with recurrence: n*a(n) +(n-1)*a(n-1) +(24-17*n)*a(n-2) +(41-17*n)*a(n-3) +72*(n-3)*a(n-4) +72*(n-4)*a(n-5)=0. - R. J. Mathar, Nov 17 2011

G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x) + 9*x^2*A(x)^2). - Paul D. Hanna, Nov 18 2014

Let G(x) = g.f. of A200375, then g.f. A(x) satisfies:

(1) A(x) = x/Series_Reversion(x*G(x)),

(2) A(x) = G(x/A(x)) and G(x) = A(x*G(x)),

where A200375(n) = A000108(n)*A001045(n), the product of Catalan and Jacobsthal numbers.

a(n) ~ 3^(n-1). - Vaclav Kotesovec, Jun 29 2013