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%I M3669 N1495 #80 Feb 16 2020 10:51:38
%S 1,4,37,559,11776,318511,10522639,410701432,18492087079,943507142461,
%T 53798399207356,3390242657205889,233980541746413697,
%U 17551930873638233164,1421940381306443299981,123726365104534205331511,11507973895102987539130504
%N Bessel polynomial y_n(3).
%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 Gheorghe Coserea and T. D. Noe, <a href="/A001518/b001518.txt">Table of n, a(n) for n = 0..200</a> (terms up to n=100 by T. D. Noe)
%H W. Mlotkowski, A. Romanowicz, <a href="http://www.math.uni.wroc.pl/~pms/files/33.2/Article/33.2.19.pdf">A family of sequences of binomial type, Probability and Mathematical Statistics</a>, Vol. 33, Fasc. 2 (2013), pp. 401-408.
%H Simon Plouffe, <a href="http://arxiv.org/abs/0911.4975">Approximations of generating functions and a few conjectures</a>, arXiv:0911.4975 [math.NT], 2009.
%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 y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!).
%F D-finite with recurrence a(n) = 3(2n-1)*a(n-1) + a(n-2). - _T. D. Noe_, Oct 26 2006
%F G.f.: 1/Q(0), where Q(k)= 1 - x - 3*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 17 2013
%F a(n) = exp(1/3)*sqrt(2/(3*Pi))*BesselK(1/2+n,1/3). - _Gerry Martens_, Jul 22 2015
%F a(n) ~ sqrt(2) * 6^n * n^n / exp(n-1/3). - _Vaclav Kotesovec_, Jul 22 2015
%F E.g.f.: exp(1/3 - 1/3*(1-6*x)^(1/2)) / (1-6*x)^(1/2). (formula due to B. Salvy, see Plouffe link) - _Gheorghe Coserea_, Aug 06 2015
%F From _G. C. Greubel_, Aug 16 2017: (Start)
%F a(n) = (1/2)_{n} * 6^n * hypergeometric1f1(-n; -2*n; 2/3).
%F G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 6*t/(1-t)^2). (End)
%p f:= gfun:-rectoproc({a(n)=3*(2*n-1)*a(n-1)+a(n-2),a(0)=1,a(1)=4},a(n),remember):
%p map(f, [$0..60]); # _Robert Israel_, Aug 06 2015
%t Table[Sum[(n+k)!*3^k/(2^k*(n-k)!*k!), {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Jul 22 2015 *)
%o (PARI) x='x+O('x^33); Vec(serlaplace(exp(1/3 - 1/3 * (1-6*x)^(1/2)) / (1-6*x)^(1/2))) \\ _Gheorghe Coserea_, Aug 04 2015
%Y Cf. A001515, A001517.
%Y Polynomial coefficients are in A001498.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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