OFFSET
0,2
COMMENTS
Let S(i,j,n) denote a sequence of the form a(0) = 1, a(1) = i, a(n) = i*n*a(n-1) + j*a(n-2). Then S(i,j,n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)*binomial(n-k,k)*i^(n-2*k)*j^k.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..380
FORMULA
a(n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)*binomial(n-k,k)*3^(n-2*k).
a(n) ~ BesselI(0, 2/3) * n! * 3^n. - Vaclav Kotesovec, Jun 03 2018
MAPLE
a:=proc(n) option remember: if n=0 then 1 elif n=1 then 3 elif n>=2 then 3*n*procname(n-1)-procname(n-2) fi; end:
seq(a(n), n=0..20); # Muniru A Asiru, Jun 01 2018
MATHEMATICA
RecurrenceTable[{a[0]==1, a[1]==3, a[n]==3n a[n-1]+a[n-2]}, a, {n, 20}] (* Harvey P. Dale, Aug 27 2019 *)
PROG
(PARI) {a(n) = sum(k=0, n/2, ((n-k)!/k!)*binomial(n-k, k)*3^(n-2*k))}
(GAP) List([0..20], n->Sum([0..Int(n/2)], k->((Factorial(n-k))/(Factorial(k))*Binomial(n-k, k)*3^(n-2*k)))); # Muniru A Asiru, Jun 01 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 01 2018
STATUS
approved