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A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.

%I A001517 M3062 N1240 #111 Oct 27 2023 19:35:43
%S A001517 1,3,19,193,2721,49171,1084483,28245729,848456353,28875761731,
%T A001517 1098127402131,46150226651233,2124008553358849,106246577894593683,
%U A001517 5739439214861417731,332993721039856822081,20651350143685984386753
%N A001517 Bessel polynomials y_n(x) (see A001498) evaluated at 2.
%C A001517 Numerators of successive convergents to e using continued fraction 1 + 2/(1 + 1/(6 + 1/(10 + 1/(14 + 1/(18 + 1/(22 + 1/26 + ...)))))).
%C A001517 Number of ways to use the elements of {1,...,k}, n <= k <= 2n, once each to form a collection of n lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005, Jun 26 2006
%D A001517 L. Euler, 1737.
%D A001517 I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed., Section 0.126, p. 2.
%D A001517 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%D A001517 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001517 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001517 Robert Israel, <a href="/A001517/b001517.txt">Table of n, a(n) for n = 0..334</a> (first 101 terms from T. D. Noe)
%H A001517 P. Bala, <a href="/A001517/a001517.pdf">A note on the Catalan transform of a sequence</a>.
%H A001517 J. W. L. Glaisher, <a href="https://archive.org/stream/reportofbritisha72brit#page/n345/mode/2up">On Lambert's proof of the irrationality of Pi and on the irrationality of certain other quantities</a>, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.
%H A001517 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=131">Encyclopedia of Combinatorial Structures 131</a>.
%H A001517 D. H. Lehmer, <a href="http://www.jstor.org/stable/1968107">Arithmetical periodicities of Bessel functions</a>, Annals of Mathematics, 33 (1932): 143-150. The sequence is on page 149.
%H A001517 D. H. Lehmer, <a href="http://dx.doi.org/10.1090/S0025-5718-46-99625-1">Review of various tables by P. Pederson</a>, Math. Comp., 2 (1946), 68-69.
%H A001517 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</a>, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
%H A001517 R. A. Proctor, <a href="https://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way for Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.
%H A001517 J. Riordan, <a href="/A001519/a001519_1.pdf">Letter to N. J. A. Sloane, Jul. 1968</a>.
%H A001517 J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>.
%H A001517 N. J. A. Sloane, <a href="/A001514/a001514.pdf">Letter to J. Riordan, Nov. 1970</a>.
%H A001517 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H A001517 <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F A001517 a(n) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!) = (e/Pi)^(1/2) K_{n+1/2}(1/2).
%F A001517 D-finite with recurrence a(n) = (4*n-2)*a(n-1) + a(n-2), n >= 2.
%F A001517 a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n+k)*binomial(n,k)*A000522(n+k). - _Vladeta Jovovic_, Sep 30 2006
%F A001517 E.g.f. (for offset 1): exp(x*c(x)), where c(x)=(1-sqrt(1-4*x))/(2*x) (cf. A000108). - _Vladimir Kruchinin_, Aug 10 2010
%F A001517 G.f.: 1/Q(0), where Q(k) = 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 17 2013
%F A001517 a(n) = (1/n!)*Integral_{x>=0} (x*(1 + x))^n*exp(-x) dx. Expansion of exp(x) in powers of y = x*(1 - x): exp(x) = 1 + y + 3*y^2/2! + 19*y^3/3! + 193*y^4/4! + 2721*y^5/5! + .... - _Peter Bala_, Dec 15 2013
%F A001517 a(n) = exp(1/2) / sqrt(Pi) * BesselK(n+1/2, 1/2). - _Vaclav Kotesovec_, Mar 15 2014
%F A001517 a(n) ~ 2^(2*n+1/2) * n^n / exp(n-1/2). - _Vaclav Kotesovec_, Mar 15 2014
%F A001517 a(n) = hypergeom([-n, n+1], [], -1). - _Peter Luschny_, Oct 17 2014
%F A001517 From _G. C. Greubel_, Aug 16 2017: (Start)
%F A001517 a(n) = (1/2)_{n} * 4^n * hypergeometric1f1(-n; -2*n; 1).
%F A001517 G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 4*t/(1-t)^2). (End)
%F A001517 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*k!. - _Ilya Gutkovskiy_, Nov 24 2017
%F A001517 a(n) = KummerU(-n, -2*n, 1). - _Peter Luschny_, May 10 2022
%p A001517 A:= gfun:-rectoproc({a(n) = (4*n-2)*a(n-1) + a(n-2),a(0)=1,a(1)=3},a(n),remember):
%p A001517 map(A, [$0..20]); # _Robert Israel_, Jul 22 2015
%p A001517 f:=proc(n) option remember; if n = 0 then 1 elif n=1 then 3 else f(n-2)+(4*n-2)*f(n-1); fi; end;
%p A001517 [seq(f(n), n=0..20)]; # _N. J. A. Sloane_, May 09 2016
%p A001517 seq(simplify(KummerU(-n, -2*n, 1)), n = 0..16); # _Peter Luschny_, May 10 2022
%t A001517 Table[(2k)! Hypergeometric1F1[-k, -2k, 1]/k!, {k, 0, 10}] (* _Vladimir Reshetnikov_, Feb 16 2011 *)
%o A001517 (PARI) a(n)=sum(k=0,n,(n+k)!/k!/(n-k)!)
%o A001517 (Sage)
%o A001517 A001517 = lambda n: hypergeometric([-n, n+1], [], -1)
%o A001517 [simplify(A001517(n)) for n in (0..16)] # _Peter Luschny_, Oct 17 2014
%Y A001517 Essentially the same as A080893.
%Y A001517 a(n) = A099022(n)/n!.
%Y A001517 Partial sums: A105747.
%Y A001517 Replace "lists" with "sets" in comment: A001515.
%Y A001517 Cf. A001515, A001518, A002119, A053556, A053557, A243593.
%K A001517 nonn,easy,nice
%O A001517 0,2
%A A001517 _N. J. A. Sloane_
%E A001517 More terms from _Vladeta Jovovic_, Apr 03 2000
%E A001517 Additional comments from _Michael Somos_, Jul 15 2002