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%I #15 Aug 04 2014 22:54:51
%S 1,1,3,18,146,1530,19620,297360,5201784,103146120,2286181800,
%T 56011087440,1503057473280,43844234353920,1381310964633600,
%U 46743301840435200,1690919874777893760,65116170597269151360,2659604669692822051200,114838104572526535200000,5226654647185285702752000
%N G.f.: Sum_{n>=0} n^n * x^n * (1 + n*x)^n / (1 + n*x + n^2*x^2)^(n+1).
%C a(n) is divisible by [n/2]!.
%C Compare definition to the following identity, which holds for all fixed k:
%C Sum_{n>=0} n!*x^n = Sum_{n>=0} x^n * (n + k*x)^n / (1 + n*x + k*x^2)^(n+1).
%F a(n) = Sum_{k=0..[n/2]} (n-k)! * Stirling2(n, n-k) * binomial(n-k,k).
%F a(n) ~ c * d^n * n^n / exp(n), where d = r*(-1+3*r-3*r^2)/(1-3*r+2*r^2) = 2.334305682349197638435662..., where r = 0.722795640379451585372396... is the root of the equation (1-r) * (r + 1/LambertW(-exp(-1/r)/r)) + (2*r-1)^2 = 0, and c = 1.04764685950245700560418116727397... . - _Vaclav Kotesovec_, Aug 03 2014
%e G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 146*x^4 + 1530*x^5 + 19620*x^6 +...
%e where
%e A(x) = 1 + x*(1+x)/(1+x+x^2)^2 + 2^2*x^2*(1+2*x)^2/(1+2*x+4*x^2)^3 + 3^3*x^3*(1+3*x)^3/(1+3*x+9*x^2)^4 + 4^4*x^4*(1+4*x)^4/(1+4*x+16*x^2)^5 +...
%t Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k,k],{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Aug 03 2014 *)
%o (PARI) /* From definition: */
%o {a(n)=local(A=1); A=sum(m=0, n, m^m*x^m*(1+m*x)^m/(1 + m*x + m^2*x^2 +x*O(x^n))^(m+1)); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* From 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)=sum(k=0,n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k,k))}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A240172, A240956, A240957, A240958, A240921, A008277 (Stirling2).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 02 2014
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