%I M4295 N1795 #39 Feb 16 2020 10:51:19
%S 0,0,6,120,1980,32970,584430,11204676,233098740,5254404210,
%T 127921380840,3350718545460,94062457204716,2819367702529560,
%U 89912640142178490,3040986592542420060,108752084073199561140,4101112025363285051526
%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="/A001516/b001516.txt">Table of n, a(n) for n = 0..400</a>
%H J. Riordan, <a href="/A001519/a001519_1.pdf">Letter to N. J. A. Sloane, Jul. 1968</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 G.f.: 6*x^2*(1-x)^(-5)*hypergeom([5/2,3],[],2*x/(x-1)^2). - Mark van Hoeij, Nov 07 2011
%F D-finite with recurrence: (n-2)*(n-1)*a(n) = (2*n - 1)*(n^2 - n + 2)*a(n-1) + n*(n+1)*a(n-2). - _Vaclav Kotesovec_, Jul 22 2015
%F a(n) ~ 2^(n+1/2) * n^(n+2) / exp(n-1). - _Vaclav Kotesovec_, Jul 22 2015
%F a(n) = n*(n - 1)*(1/2)_{n}*2^n* hypergeometric1F1(2 - n, -2*n, 2), where (a)_{n} is the Pochhammer symbol. - _G. C. Greubel_, Aug 14 2017
%F E.g.f.: (-1)*(1 - 2*x)^(-5/2)*((4 - 14*x + 9*x^2)*sqrt(1 - 2*x) + (2*x^3 - 24*x^2 + 18*x - 4))*exp((1 - sqrt(1 - 2*x))). - _G. C. Greubel_, Aug 16 2017
%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$2)),n=0..60)];
%t Table[Sum[(n+k+2)!/(2^(k+2)*(n-k-2)!*k!), {k,0,n-2}], {n,0,20}] (* _Vaclav Kotesovec_, Jul 22 2015 *)
%t Join[{0, 0}, Table[n*(n - 1)*Pochhammer[1/2, n]*2^n* Hypergeometric1F1[2 - n, -2*n, 2], {n,2,50}]] (* _G. C. Greubel_, Aug 14 2017 *)
%o (PARI) for(n=0,20, print1(sum(k=0,n-2, (n+k+2)!/(2^(k+2)*(n-k-2)!*k!)), ", ")) \\ _G. C. Greubel_, Aug 14 2017
%Y Cf. A001497, A001498, A001514, A001515, A001518, A065944, A144505.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_