OFFSET
0,2
COMMENTS
The number of 3-level labeled linear rooted trees with 2*n leaves.
A bisection of A050351.
a(n) == 5 (mod 360) for n>0.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..195
FORMULA
E.g.f.: 1/2 + (1/6)*Sum_{n>=0} exp(n^2*x) * (2/3)^n = Sum_{n>=0} a(n)*x^n/n!.
a(n) = Sum_{k=0..2*n} 2^(k-1) * k! * Stirling2(2*n, k) for n>0 with a(0)=1. [After Vladeta Jovovic in A050351]
a(n) ~ (2*n)! / (6 * (log(3/2))^(2*n+1)). - Vaclav Kotesovec, Nov 29 2014
EXAMPLE
E.g.f.: E(x) = 1 + 5*x^2/2! + 365*x^4/4! + 66605*x^6/6! + 22687565*x^8/8! +...
where E(x) = (8 - 7*cosh(x)) / (13 - 12*cosh(x)), or, equivalently,
E(x) = (7 - 16*exp(x) + 7*exp(2*x)) / (12 - 26*exp(x) + 12*exp(2*x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 5*x + 365*x^2/2! + 66605*x^3/3! + 22687565*x^4/4! +...
where
6*A(x) = 4 + exp(x)*(2/3) + exp(4*x)*(2/3)^2 + exp(9*x)*(2/3)^3 + exp(16*x)*(2/3)^4 + exp(25*x)*(2/3)^5 + exp(36*x)*(2/3)^6 + exp(49*x)*(2/3)^7 +...
MATHEMATICA
nmax=20; Table[(CoefficientList[Series[(8-7*Cosh[x]) / (13-12*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[n]], {n, 1, 2*nmax+2, 2}] (* Vaclav Kotesovec, Nov 29 2014 *)
PROG
(PARI) /* E.g.f.: (8 - 7*cosh(x)) / (13 - 12*cosh(x)): */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (8 - 7*cosh(X)) / (13 - 12*cosh(X)) , 2*n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=0, 2*n, 2^(k-1) * k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* As the Sum of an Infinite Series: */
\p60 \\ set precision
Vec(serlaplace(1/2+1/6*sum(n=0, 2000, exp(n^2*x)*(2/3)^n*1.)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 28 2014
STATUS
approved