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%I #19 Aug 06 2014 06:02:24
%S 1,2,9,72,830,12420,228060,4959360,124589304,3550050000,113116311000,
%T 3985174226880,153815533185600,6454433029524480,292557975636326400,
%U 14244829479956275200,741502151945703308160,41092028680670274827520,2415394879269218890243200
%N G.f.: Sum_{n>=0} n^n * x^n * (2 + n*x)^n / (1 + n*x)^(2*n+2).
%H Vincenzo Librandi, <a href="/A240956/b240956.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..[n/2]} (n-k)! * Stirling2(n, n-k) * binomial(n-k, k) * 2^(n-2*k).
%F a(n) ~ c * d^n * n! / sqrt(n), where d = 2*r^2/(2*r-1) + (2*r-1)*r/(2*(1-r)) = 3.36074272900128370245729732045120604190737486342012..., where r = 0.80276231206743119172295651485200150958072822575039811732... is the root of the equation (r + (1-2*r)^2/(4*(1-r))) * LambertW(-exp(-1/r)/r) = -1, and c = 0.533888836381702228067397487907088688592161798613354080016... . - _Vaclav Kotesovec_, Aug 05 2014
%e O.g.f.: A(x) = 1 + 2*x + 9*x^2 + 72*x^3 + 830*x^4 + 12420*x^5 + 228060*x^6 +...
%e where
%e A(x) = 1 + x*(2+x)/(1+x)^4 + 2^2*x^2*(2+2*x)^2/(1+2*x)^6 + 3^3*x^3*(2+3*x)^3/(1+3*x)^8 + 4^4*x^4*(2+4*x)^4/(1+4*x)^10 + 5^5*x^5*(2+5*x)^5/(1+5*x)^12 +...
%t Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k, k] * 2^(n-2*k), {k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Aug 05 2014 *)
%o (PARI) {a(n)=local(A=1); A=sum(m=0, n, m^m*x^m*(2+m*x)^m/(1 + m*x +x*O(x^n))^(2*m+2)); 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)*2^(n-2*k) )}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A240957, A240958.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 04 2014
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