%I #36 Sep 12 2021 15:27:15
%S 1,2,21,555,27930,2297295,280737765,47773195470,10803712844925,
%T 3134455177028175,1135027918156081950,501797959522466381775,
%U 265999339071854103540825,166538120746634232882536250,121585839578169857291258983125,102384090433785464586295830691875
%N a(n+2) = (2*n+1)^2*a(n+1) + (2*n+1)*(2*n-1)*a(n) with a(1)=1 and a(2)=2.
%H Harvey P. Dale, <a href="/A218768/b218768.txt">Table of n, a(n) for n = 1..226</a>
%F a(n) = (e/Pi)*Integral_{z=-infinity..infinity} z^(2*n) K(0, sqrt(1+z^2)), where K(0, x) is the modified Bessel function of the second kind with order 0 (empirical observation).
%F a(1) = 1, a(2) = 2, a(n) = (2*n-3)^2*a(n-1) + (2*n-3)*(2*n-5)*a(n-2) for n > 2. - _Andrew Howroyd_, Dec 24 2019
%t Table[DifferenceRoot[Function[{f,k}, {f[k+2]==(2k+1)^2 f[k+1]+(2k+1)(2k-1)f[k], f[1]==1, f[2]==2}]][n], {n,1,15}] (* Corrected by _Wesley Transue_, Dec 23 2019 *)
%t Nest[Append[#1, (2 #2 + 1)^2*#1[[#2 + 1]] + (2 #2 + 1) (2 #2 - 1) #1[[-2]]] & @@ {#, Length@ # - 1} &, {1, 2}, 14] (* _Michael De Vlieger_, Dec 24 2019 *)
%t RecurrenceTable[{a[1]==1,a[2]==2,a[n+2]==(2n+1)^2 a[n+1]+(2n+1)(2n-1) a[n]},a,{n,20}] (* _Harvey P. Dale_, Sep 12 2021 *)
%o (PARI) seq(n)={my(a=vector(n)); a[1]=1; a[2]=2; for(n=3, #a, a[n]=(2*n-3)^2*a[n-1] + (2*n-3)*(2*n-5)*a[n-2]); a} \\ _Andrew Howroyd_, Dec 24 2019
%K easy,nonn
%O 1,2
%A _Wesley Transue_, Nov 05 2012
%E Offset corrected by _Wesley Transue_, Dec 23 2019