A216454 G.f. satisfies: A(x) = (1 + x*(3-x)*A(x)) * (1 + x^2*A(x)).

1, 3, 9, 30, 107, 396, 1503, 5820, 22907, 91371, 368523, 1500354, 6157669, 25448907, 105821865, 442409844, 1858482563, 7840766601, 33207750819, 141137445258, 601768494201, 2573246794374, 11032997250357, 47421297986868, 204286464525165, 881900059488741

Offset: 0

Author

Paul D. Hanna, Sep 10 2012

Keywords

• nonn

Comments

The radius of convergence of the g.f. A(x) is r = 2/(5+sqrt(17)) with A(r) = 2/(1-3*r) = (11+3*sqrt(17))/4.

More generally, if A(x) = (1 + x*(t-x)*A(x)) * (1 + x^2*A(x)), |t|>0, then

A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k) )

where the radius of convergence r of the g.f. A(x) satisfies

r = (1-r)^2/(t-r) = (1-t*r)/(2*(1-r)) with A(r) = 1/(r*(1-r)) = 2/(1-t*r).

Examples

			G.f.: A(x) = 1 + 3*x + 9*x^2 + 30*x^3 + 107*x^4 + 396*x^5 + 1503*x^6 +...
The logarithm of the g.f. begins:
log(A(x)) = ((3-x) + x)*x + ((3-x)^2 + 2^2*x*(3-x) + x^2)*x^2/2 +
((3-x)^3 + 3^2*x*(3-x)^2 + 3^2*x^2*(3-x) + x^3)*x^3/3 +
((3-x)^4 + 4^2*x*(3-x)^3 + 6^2*x^2*(3-x)^2 + 4^2*x^3*(3-x) + x^4)*x^4/4 +
((3-x)^5 + 5^2*x*(3-x)^4 + 10^2*x^2*(3-x)^3 + 10^2*x^3*(3-x)^2 + 5^2*x^4*(3-x) + x^5)*x^5/5 +...
Explicitly,
log(A(x)) = 3*x + 9*x^2/2 + 36*x^3/3 + 149*x^4/4 + 618*x^5/5 + 2592*x^6/6 + 11007*x^7/7 + 47181*x^8/8 + 203634*x^9/9 + 883674*x^10/10 +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A216604, A216434.

Programs

Mathematica

CoefficientList[Series[2/(1-3*x+Sqrt[(1-3*x)^2-4*x^3*(3-x)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 17 2013 *)

PARI

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(3-x)^(m-k) + x*O(x^n)))),n)}

PARI

{a(n)=polcoeff(2/(1-3*x+sqrt((1-3*x)^2-4*x^3*(3-x) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(3-x)^(n-k) ).

G.f.: 2 / ( 1-3*x + sqrt( (1-3*x)^2 - 4*x^3*(3-x) ) ).

Recurrence: 3*(n+3)*a(n) = (19*n+30)*a(n-1) - 3*(11*n+3)*a(n-2) + 9*(5*n-6)*a(n-3) - 6*(4*n-9)*a(n-4) + 4*(n-3)*a(n-5). - Vaclav Kotesovec, Sep 17 2013

a(n) ~ sqrt(7378+1794*sqrt(17)) * ((5+sqrt(17))/2)^n / (16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 17 2013