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%I #21 Aug 23 2017 10:50:30
%S 1,3,31,520,11991,350889,12428746,516450792,24619176153,1323971052261,
%T 79280864647205,5231080689880500,377062508515478306,
%U 29479066783583059530,2484534527715953700780,224559818606249783480400,21666961097367611148157815,2222844864226101120054773295
%N G.f.: Sum_{n>=0} exp(-(1 + n^2*x)) * (1 + n^2*x)^n / n!.
%C Compare the g.f. to:
%C (1) Sum_{n>=0} exp(-(1+n*x)) * (1+n*x)^n / n! = 1/(1-x).
%C (2) Sum_{n>=1} exp(-n^2*x) * n^(2*n) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n (A007820).
%H Paul D. Hanna and Vaclav Kotesovec, <a href="/A245109/b245109.txt">Table of n, a(n) for n = 0..200</a> (first 100 terms from Paul D. Hanna)
%F a(n) ~ c * d^n * (n-1)!, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348035823168..., and c = 10.427337127699040838035... . - _Vaclav Kotesovec_, Jul 12 2014
%F a(n) = A049020(2n,n). - _Alois P. Heinz_, Aug 23 2017
%e G.f.: A(x) = 1 + 3*x + 31*x^2 + 520*x^3 + 11991*x^4 + 350889*x^5 +...
%e where
%e A(x) = exp(-1) + exp(-(1+x))*(1+x) + exp(-(1+2^2*x))*(1+2^2*x)^2/2!
%e + exp(-(1+3^2*x))*(1+3^2*x)^3/3! + exp(-(1+4^2*x))*(1+4^2*x)^4/4!
%e + exp(-(1+5^2*x))*(1+5^2*x)^5/5! + exp(-(1+6^2*x))*(1+6^2*x)^6/6!
%e + exp(-(1+7^2*x))*(1+7^2*x)^7/7! + exp(-(1+8^2*x))*(1+8^2*x)^8/8! +...
%e simplifies to a power series in x with integer coefficients.
%t Table[SeriesCoefficient[Sum[E^(-(1+k^2*x))*(1+k^2*x)^k/k!,{k,0,Infinity}],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 12 2014 *)
%o (PARI) /* Must first set suitable precision */ \p300
%o {a(n)=local(A=1+x); A=suminf(k=0, exp(-(1+k^2*x)+x*O(x^n))*(1+k^2*x)^k/k!); round(polcoeff(A, n))}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A049020, A245110.
%Y Cf. A187655, A217899, A217900.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 12 2014
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