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 A064570 #27 May 31 2019 17:27:33
%S A064570 1,3,29,799,43353,3837851,501393973,90608944119,21633834338609,
%T A064570 6593857931708083,2497877833687172301,1151118261673522046543,
%U A064570 634098400947597342716809,411445662820653995008883019
%N A064570 Binomial transform of (2n)!.
%C A064570 Compare with A229464. - _Peter Bala_, Sep 25 2013
%H A064570 Vincenzo Librandi, <a href="/A064570/b064570.txt">Table of n, a(n) for n = 0..100</a>
%F A064570 In Maple notation: a(n)=hypergeom([1, 1/2, -n], [], -4), n=0, 1, ...
%F A064570 a(n) = Integral_{x>=0} ((x^4-1)/(x^2-1))^n*exp(-x) dx. - _Gerald McGarvey_, Oct 14 2006
%F A064570 From _Peter Bala_, Sep 25 2013: (Start)
%F A064570 a(n) = Sum_{k = 0..n} binomial(n,k)*(2*k)!.
%F A064570 Clearly a(n) is always odd; indeed, a(n) = 1 + 2*n*A229464(n-1) for n >= 1.
%F A064570 Recurrence equation: a(n) = 1 + 2*n*(2*n - 1)*a(n-1) - 2*n*(2*n - 2)*a(n-2) with a(0) = 1 and a(1) = 3.
%F A064570 O.g.f. Sum_{k >= 0} (2*k)!*x^k/(1 - x)^(k + 1) = 1 + 3*x + 29*x^2 + 799*x^3 + .... (End)
%F A064570 Recurrence (homogeneous): a(n) = (4*n^2 - 2*n + 1)*a(n-1) - 2*(n-1)*(4*n-3)*a(n-2) + 4*(n-2)*(n-1)*a(n-3). - _Vaclav Kotesovec_, Sep 26 2013
%F A064570 a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / exp(2*n). - _Vaclav Kotesovec_, Sep 26 2013
%F A064570 From _Peter Bala_, Nov 26 2017: (Start)
%F A064570 E.g.f.: exp(x)*Sum_{n >= 0} A001813(n)*x^n.
%F A064570 a(k) = a(0) (mod k) for all k (by the inhomogeneous recurrence equation).
%F A064570 More generally a(n+k) = a(n) (mod k) for all n and k by an induction argument on n.
%F A064570 It follows that for each positive integer k, the sequence a(n) (mod k) is periodic, with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 3, 9, 9, 3, 1, 3, 9, 9, 3, ... with exact period 5. (End)
%t A064570 Table[Sum[Binomial[n,k]*(2*k)!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 26 2013 *)
%Y A064570 Cf. A010050, A229464, A001813, A323280.
%K A064570 nonn,easy
%O A064570 0,2
%A A064570 _Karol A. Penson_, Sep 20 2001