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A250915
E.g.f.: (32 - 31*cosh(x)) / (41 - 40*cosh(x)).
5
1, 9, 2169, 1306809, 1469709369, 2656472295609, 7042235448544569, 25740278881968596409, 124066865052334175027769, 762445058190042799428289209, 5818666543923901596429593478969, 53987940899344324456042542132654009, 598504142090716188282023260396781018169
OFFSET
0,2
COMMENTS
The number of 5-level labeled linear rooted trees with 2*n leaves.
A bisection of A050353.
a(n) == 9 (mod 2160) for n>0.
LINKS
FORMULA
E.g.f.: 3/4 + (1/20)*Sum_{n>=0} exp(n^2*x) * (4/5)^n = Sum_{n>=0} a(n)*x^n/n!.
a(n) = Sum_{k=0..2*n} 4^(k-1) * k! * Stirling2(2*n, k) for n>0 with a(0)=1.
a(n) ~ (2*n)! / (20 * (log(5/4))^(2*n+1)). - Vaclav Kotesovec, Nov 29 2014
EXAMPLE
E.g.f.: E(x) = 1 + 9*x^2/2! + 2169*x^4/4! + 1306809*x^6/6! + 1469709369*x^8/8! +...
where E(x) = (32 - 31*cosh(x)) / (41 - 40*cosh(x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 9*x + 2169*x^2/2! + 1306809*x^3/3! + 1469709369*x^4/4! +...
where
20*A(x) = 16 + exp(x)*(4/5) + exp(4*x)*(4/5)^2 + exp(9*x)*(4/5)^3 + exp(16*x)*(4/5)^4 + exp(25*x)*(4/5)^5 + exp(36*x)*(4/5)^6 +...
MATHEMATICA
nmax=20; Table[(CoefficientList[Series[(32-31*Cosh[x]) / (41-40*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.: (32 - 31*cosh(x)) / (41 - 40*cosh(x)): */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (32 - 31*cosh(X)) / (41 - 40*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, 4^(k-1) * k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* As the Sum of an Infinite Series: */
\p100 \\ set precision
Vec(serlaplace(3/4 + 1/20*sum(n=0, 3000, exp(n^2*x)*(4/5)^n*1.)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 28 2014
STATUS
approved