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%I #22 Sep 18 2014 11:07:58
%S 1,3,20,188,2214,30922,495816,8931960,177999366,3878476418,
%T 91558971096,2324529942088,63084714688540,1820757355281828,
%U 55645592361311504,1794034726184859120,60817844748284215110,2161623389394872099250,80346564637248427227960
%N Bateman polynomial values n!*Z_n(-1).
%H Vincenzo Librandi, <a href="/A073767/b073767.txt">Table of n, a(n) for n = 0..200</a>
%H M. C. Fasenmyer, <a href="http://www.jstor.org/stable/2305810">A note on pure recurrence relations</a>, Amer. Math. Monthly 56, (1949), 14-17. Math. Rev. 10,704b.
%F a(n) = n!(Sum k=0..n (n+k)!/(k!^3(n-k)!)) = n!*F(-n, n+1;1, 1;-1).
%F n(2n-3)a(n) = (2n-1)(3n^2-2n-4)a(n-1)-(2n-3)(3n^2-10n+4)(n-1)a(n-2)+(n-1)(2n-1)(n-2)^3a(n-3).
%F E.g.f.: exp(2*x/(x-1)^2)*BesselI(0,2*x/(x-1)^2)/(1-x). - _Mark van Hoeij_, Oct 24 2011
%F a(n) ~ n^(n-1/6) * exp(3*n^(2/3)-n-1/3) / sqrt(6*Pi) * (1 + 1/n^(1/3) + 61/(90*n^(2/3))). - _Vaclav Kotesovec_, Feb 25 2014
%t Table[n!*Sum[(n+k)!/(n-k)!/k!^3,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Feb 25 2014 *)
%o (PARI) a(n)=if(n<0,0,n!*sum(k=0,n,(n+k)!/(n-k)!/k!^3))
%o (Sage)
%o A073767 = lambda n: factorial(n)*hypergeometric([-n,n+1], [1,1], -1)
%o [round(A073767(n).n(100)) for n in (0..18)] # _Peter Luschny_, Sep 18 2014
%Y Cf. A073768.
%K nonn
%O 0,2
%A _Michael Somos_, Aug 08 2002
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