A305460 a(0) = 1, a(1) = 3, a(n) = 3*n*a(n-1) + 2*a(n-2).

1, 3, 20, 186, 2272, 34452, 624680, 13187184, 317741776, 8605402320, 258797553152, 8557530058656, 308588677217920, 12052073471616192, 506804263162315904, 22830295989247448064, 1096867816010202138880, 55985919208498803979008

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. A222467, A305459.

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)*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)+2*procname(n-2) fi; end:
seq(a(n),n=0..20); # Muniru A Asiru, Jun 01 2018

Mathematica

nxt[{n_,a_,b_}]:={n+1,b,3b(n+1)+2a}; NestList[nxt,{1,1,3},20][[;;,2]] (* Harvey P. Dale, Feb 08 2025 *)

PARI

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

Formula

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

a(n) ~ BesselI(0, 2*sqrt(2)/3) * n! * 3^n. - Vaclav Kotesovec, Jun 03 2018