OFFSET
0,2
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..658
Magdalena Boos, Giovanni Cerulli Irelli, Francesco Esposito, Parabolic orbits of 2-nilpotent elements for classical groups, arXiv:1802.06425 [math.RT], 2018.
FORMULA
a(n) = Sum_{k=0..[n/2]} 3^(n-2*k)*2^k * n!/((n-2*k)!*k!).
O.g.f.: 1/(1-3*x - 4*x^2/(1-3*x - 8*x^2/(1-3*x - 12*x^2/(1-3*x - 16*x^2/(1-3*x -...))))), a continued fraction.
Recurrence: a(n) = 3*a(n-1) + 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 2^(n-1/2)*exp(3/2*sqrt(n)-n/2-9/16)*n^(n/2)*(1+33/(64*sqrt(n))). - Vaclav Kotesovec, Oct 20 2012
EXAMPLE
E.g.f.: 1 + 3*x + 13*x^2/2! + 63*x^3/3! + 345*x^4/4! + 2043*x^5/5! +...
MATHEMATICA
CoefficientList[Series[E^(3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)
PROG
(PARI) {a(n)=n!*polcoeff(exp(3*x+2*x^2+x*O(x^n)), n)}
(PARI) {a(n)=sum(k=0, n\2, 3^(n-2*k)*2^k*n!/((n-2*k)!*k!))}
(PARI) /* O.g.f. as a continued fraction: */
{a(n)=local(CF=1+3*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-3*x-4*(n-k)*x^2*CF)); polcoeff(CF, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 25 2011
STATUS
approved