A305459 a(0) = 1, a(1) = 3, a(n) = 3*n*a(n-1) + a(n-2).

1, 3, 19, 174, 2107, 31779, 574129, 12088488, 290697841, 7860930195, 236118603691, 7799774851998, 281028013275619, 10967892292601139, 460932504302523457, 20752930585906156704, 996601600627798045249, 50847434562603606464403

Offset: 0

Author

Seiichi Manyama, Jun 01 2018

Keywords

• nonn

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

Crossrefs

Cf. A001040, A036243, A213190, A305460.

Programs

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

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 *)

PARI

{a(n) = sum(k=0, n/2, ((n-k)!/k!)*binomial(n-k,k)*3^(n-2*k))}

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