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A102757 a(n) = Sum_{i=0..n} C(n,i)^2 * i! * 3^i.

%I A102757 #21 Feb 02 2021 11:55:42
%S A102757 1,4,31,352,5233,95836,2080999,52189096,1482977857,47053929268,
%T A102757 1648037039791,63125834205424,2624096058047281,117620219281363852,
%U A102757 5653607876781921463,290035426344483253816,15814774125898034896129
%N A102757 a(n) = Sum_{i=0..n} C(n,i)^2 * i! * 3^i.
%C A102757 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 A102757 Seiichi Manyama, <a href="/A102757/b102757.txt">Table of n, a(n) for n = 0..377</a>
%F A102757 E.g.f.: 1/(1-3x)*exp(x/(1-3x)).
%F A102757 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 A102757 a(n) = 2*(3*n-1)*a(n-1) - 9*(n-1)^2*a(n-2). - _Vaclav Kotesovec_, Sep 29 2013
%F A102757 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 A102757 seq(sum('binomial(k,i)^2*i!*3^i', 'i'=0..k),k=0..30);
%t A102757 f[n_] := Sum[k!*3^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 16}] (* or *)
%t A102757 Range[0, 16]! CoefficientList[ Series[1/(1 - 3x)*Exp[x/(1 - 3x)], {x, 0, 16}], x] (* _Robert G. Wilson v_, Mar 16 2005 *)
%Y A102757 Cf. A002720, A025167, A102773.
%K A102757 easy,nonn
%O A102757 0,2
%A A102757 _Miklos Kristof_, Mar 16 2005
%E A102757 More terms from _Robert G. Wilson v_, Mar 16 2005