cp's OEIS Frontend

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

%I A305460 #19 Feb 08 2025 14:40:52
%S A305460 1,3,20,186,2272,34452,624680,13187184,317741776,8605402320,
%T A305460 258797553152,8557530058656,308588677217920,12052073471616192,
%U A305460 506804263162315904,22830295989247448064,1096867816010202138880,55985919208498803979008
%N A305460 a(0) = 1, a(1) = 3, a(n) = 3*n*a(n-1) + 2*a(n-2).
%C A305460 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.
%H A305460 Seiichi Manyama, <a href="/A305460/b305460.txt">Table of n, a(n) for n = 0..380</a>
%F A305460 a(n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)*binomial(n-k,k)*3^(n-2*k)*2^k.
%F A305460 a(n) ~ BesselI(0, 2*sqrt(2)/3) * n! * 3^n. - _Vaclav Kotesovec_, Jun 03 2018
%p A305460 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:
%p A305460 seq(a(n),n=0..20); # _Muniru A Asiru_, Jun 01 2018
%t A305460 nxt[{n_,a_,b_}]:={n+1,b,3b(n+1)+2a}; NestList[nxt,{1,1,3},20][[;;,2]] (* _Harvey P. Dale_, Feb 08 2025 *)
%o A305460 (PARI) {a(n) = sum(k=0, n/2, ((n-k)!/k!)*binomial(n-k, k)*3^(n-2*k)*2^k)}
%o A305460 (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
%Y A305460 Cf. A222467, A305459.
%K A305460 nonn
%O A305460 0,2
%A A305460 _Seiichi Manyama_, Jun 01 2018