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%I #15 Aug 06 2014 15:45:39
%S 1,1,4,30,296,3840,60480,1127280,24240384,590728320,16090099200,
%T 484387706880,15971308784640,572403619307520,22155942961152000,
%U 921115890645350400,40935834850710159360,1936630231160472207360,97172886828612884889600,5154401709528015200256000
%N G.f.: Sum_{n>=0} n^n * x^n * (1 + 2*n*x)^n / (1 + n*x + 2*n^2*x^2)^(n+1).
%C Generally, if Sum_{n>=0} n^n * x^n * (s + t*n*x)^n / (1 + s*n*x + t*n^2*x^2)^(n+1) = Sum_{n>=0} b(n)*x^n,
%C then b(n) = Sum_{k=0..n} (n-k)! * Stirling2(n, n-k) * binomial(n-k, k) * s^(n-2*k) * t^k.
%C Limit n->infinity (b(n)/n!)^(1/n) = s*r^2/(2*r-1) + (2*r-1)*r*t/((1-r)*s), where r is the root of the equation (r + (1-2*r)^2 * t/((1-r)*s^2)) * LambertW(-exp(-1/r)/r) = -1. - _Vaclav Kotesovec_, Aug 05 2014
%H Vincenzo Librandi, <a href="/A240958/b240958.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..n} (n-k)! * Stirling2(n, n-k) * binomial(n-k, k) * 2^k.
%F a(n) ~ c * d^n * n! / sqrt(n), where d = r^2/(2*r-1) + 2*(2*r-1)*r/(1-r) = 2.8672948250470036038473588196568091418984738141..., where r = 0.6842203847910787866923284795680321317882484098... is the root of the equation (r + 2*(1-2*r)^2/(1-r)) * LambertW(-exp(-1/r)/r) = -1, and c = 0.37767441309257908887250708986031213641309631613... . - _Vaclav Kotesovec_, Aug 05 2014
%e O.g.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 296*x^4 + 3840*x^5 + 60480*x^6 +...
%e where
%e A(x) = 1 + x*(1+2*x)/(1+x+2*x^2)^2 + 2^2*x^2*(1+4*x)^2/(1+2*x+8*x^2)^3 + 3^3*x^3*(1+6*x)^3/(1+3*x+18*x^2)^4 + 4^4*x^4*(1+8*x)^4/(1+4*x+32*x^2)^5 + 5^5*x^5*(1+10*x)^5/(1+5*x+50*x^2)^6 +...
%t Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k, k] * 2^k, {k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Aug 05 2014 *)
%o (PARI) /* By a general formula for o.g.f.: */
%o {a(n,s,t)=local(A=1); A=sum(m=0, n, m^m*x^m*(s + t*m*x)^m/(1 + s*m*x + t*m^2*x^2 +x*O(x^n))^(m+1)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n,1,2), ", "))
%o (PARI) /* By a general formula for a(n): */
%o {Stirling2(n, k)=sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!}
%o {a(n,s,t)=sum(k=0, n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k, k)*s^(n-2*k)*t^k)}
%o for(n=0, 30, print1(a(n,1,2), ", "))
%Y Cf. A240956, A240957, A240921.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 04 2014
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