This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A065919 #50 Oct 05 2023 12:20:34
%S A065919 1,5,61,1225,34361,1238221,54516085,2836074641,170218994545,
%T A065919 11577727703701,880077524475821,73938089783672665,6803184337622361001,
%U A065919 680392371852019772765,73489179344355757819621,8525425196317119926848801,1057226213522667226687070945
%N A065919 Bessel polynomial y_n(4).
%C A065919 Main diagonal of A143411. - _Peter Bala_, Aug 14 2008
%D A065919 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A065919 Harry J. Smith, <a href="/A065919/b065919.txt">Table of n, a(n) for n = 0..100</a>
%H A065919 W. Mlotkowski and 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 A065919 <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F A065919 y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!).
%F A065919 From _Peter Bala_, Aug 14 2008: (Start)
%F A065919 Recurrence relation: a(0) = 1, a(1) = 5, a(n) = 4*(2*n-1)*a(n-1) + a(n-2) for n >= 2. Sequence A143412(n) satisfies the same recurrence relation.
%F A065919 1/sqrt(e) = 1 - 2*Sum_{n = 0..inf} (-1)^n/(a(n)*a(n+1)) = 1 - 2*( 1/(1*5) - 1/(5*61) + 1/(61*1225) - ... ). (End)
%F A065919 G.f.: 1/Q(0), where Q(k)= 1 - x - 4*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 17 2013
%F A065919 a(n) = exp(1/4)/sqrt(2*Pi)*BesselK(n+1/2,1/4). - _Gerry Martens_, Jul 22 2015
%F A065919 a(n) ~ 2^(3*n+1/2) * n^n / exp(n-1/4). - _Vaclav Kotesovec_, Jul 22 2015
%F A065919 From _Peter Bala_, Apr 12 2017: (Start)
%F A065919 a(n) = 1/n!*Integral_{x = 0..inf} x^n*(1 + 2*x)^n dx.
%F A065919 E.g.f.: d/dx( exp(x*c(2*x)) ) = 1 + 5*x + 61*x^2/2! + 1225*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%F A065919 G.f.: (1/(1-x))*hypergeometric2f0(1,1/2; - ; 8*x/(1-x)^2). - _G. C. Greubel_, Aug 16 2017
%F A065919 a(n) = 2^n*KummerU(-n, -2*n, 1/2). - _Peter Luschny_, May 10 2022
%p A065919 seq(simplify(2^n*KummerU(-n,-2*n,1/2)), n=0..16); # _Peter Luschny_, May 10 2022
%t A065919 Table[Sum[(n+k)!*2^k/((n-k)!*k!), {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Jul 22 2015 *)
%o A065919 (PARI) for (n=0, 100, if (n>1, a=4*(2*n - 1)*a1 + a2; a2=a1; a1=a, if (n, a=a1=5, a=a2=1)); write("b065919.txt", n, " ", a) ) \\ _Harry J. Smith_, Nov 04 2009
%o A065919 (PARI) a(n) = sum(k=0,n, (n+k)!*2^k/((n-k)!*k!) ); \\ _Joerg Arndt_, May 17 2013
%o A065919 (Magma)
%o A065919 A065919:= func< n | (&+[Binomial(n,k)*Factorial(n+k)*2^k/Factorial(n): k in [0..n]]) >;
%o A065919 [A065919(n): n in [0..30]]; // _G. C. Greubel_, Oct 05 2023
%o A065919 (SageMath)
%o A065919 def A065919(n): return sum(binomial(n,k)*factorial(n+k)*2^k/factorial(n) for k in range(n+1))
%o A065919 [A065919(n) for n in range(31)] # _G. C. Greubel_, Oct 05 2023
%Y A065919 Cf. A001515, A001517, A001518.
%Y A065919 Cf. A143411 (main diagonal), A143412.
%Y A065919 Polynomial coefficients are in A001498.
%K A065919 nonn,easy
%O A065919 0,2
%A A065919 _N. J. A. Sloane_, Dec 08 2001