%I M4654 N1993 #37 Feb 16 2020 10:51:08
%S 0,1,9,81,835,9990,137466,2148139,37662381,733015845,15693217705,
%T 366695853876,9289111077324,253623142901401,7425873460633005,
%U 232122372003909045,7715943399320562331,271796943164015920914,10114041937573463433966
%N Bessel polynomial {y_n}'(1).
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A001514/b001514.txt">Table of n, a(n) for n = 0..400</a>
%H N. J. A. Sloane, <a href="/A001514/a001514.pdf">Letter to J. Riordan, Nov. 1970</a>
%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F a(n) = (1/2) * Sum_{k=0..n} (n+k+2)!/((n-k)!*k!*2^k) (with a different offset).
%F D-finite with recurrence: (n-1)^2 * a(n) = (2*n-1)*(n^2 - n + 1)*a(n-1) + n^2*a(n-2). - _Vaclav Kotesovec_, Jul 22 2015
%F a(n) ~ 2^(n+1/2) * n^(n+1) / exp(n-1). - _Vaclav Kotesovec_, Jul 22 2015
%F a(n) = n*2^n*(1/2)_{n}*hypergeometric1f1(1-n, -2*n, 2), where (a)_{n} is the Pochhammer symbol. - _G. C. Greubel_, Aug 14 2017
%F From _G. C. Greubel_, Aug 16 2017: (Start)
%F G.f.: (1/(1-t))*hypergeometric2f0(2, 3/2; -; 2*t/(1-t)^2).
%F E.g.f.: (1 - 2*x)^(-3/2)*((1 - x)*sqrt(1 - 2*x) + (3*x - 1))*exp((1 - sqrt(1 - 2*x))). (End)
%p (As in A001497 define:) f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
%p [seq( subs(x=1,diff(f(n),x)),n=0..60)];
%p f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..60)]; # uses a different offset
%t Table[Sum[(n+k+1)!/((n-k-1)!*k!*2^(k+1)), {k,0,n-1}], {n,0,20}] (* _Vaclav Kotesovec_, Jul 22 2015 *)
%t Join[{0}, Table[n*Pochhammer[1/2, n]*2^n* Hypergeometric1F1[1 - n, -2*n, 2], {n,1,50}]] (* _G. C. Greubel_, Aug 14 2017 *)
%o (PARI) for(n=0,50, print1(sum(k=0,n-1, (n+k+1)!/((n-k-1)!*k!*2^(k+1))), ", ")) \\ _G. C. Greubel_, Aug 14 2017
%Y Cf. A001515, A001516, A001518, A065920, A144505.
%K nonn
%O 0,3
%A _N. J. A. Sloane_