%I #21 Feb 02 2021 11:55:42

%S 1,4,31,352,5233,95836,2080999,52189096,1482977857,47053929268,

%T 1648037039791,63125834205424,2624096058047281,117620219281363852,

%U 5653607876781921463,290035426344483253816,15814774125898034896129

%N a(n) = Sum_{i=0..n} C(n,i)^2 * i! * 3^i.

%C Primes in this sequence include: a(2)=31, a(4)=5233. Semiprimes in this sequence include: a(1) = 2^2, a(6) = 31 * 67129, a(8) = 127 * 11676991. - _Jonathan Vos Post_, Mar 17 2005

%H Seiichi Manyama, <a href="/A102757/b102757.txt">Table of n, a(n) for n = 0..377</a>

%F E.g.f.: 1/(1-3x)*exp(x/(1-3x)).

%F E.g.f.: exp(3*x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. [_Paul D. Hanna_, Nov 18 2011]

%F a(n) = 2*(3*n-1)*a(n-1) - 9*(n-1)^2*a(n-2). - _Vaclav Kotesovec_, Sep 29 2013

%F a(n) ~ (3*n)^(n+1/4)*exp(2*sqrt(n/3)-n-1/6)/sqrt(2) * (1 + 103/(144*sqrt(3*n))). - _Vaclav Kotesovec_, Sep 29 2013

%p seq(sum('binomial(k,i)^2*i!*3^i', 'i'=0..k),k=0..30);

%t f[n_] := Sum[k!*3^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 16}] (* or *)

%t Range[0, 16]! CoefficientList[ Series[1/(1 - 3x)*Exp[x/(1 - 3x)], {x, 0, 16}], x] (* _Robert G. Wilson v_, Mar 16 2005 *)

%Y Cf. A002720, A025167, A102773.

%K easy,nonn

%O 0,2

%A _Miklos Kristof_, Mar 16 2005

%E More terms from _Robert G. Wilson v_, Mar 16 2005