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A229464 Binomial transform of (2*n + 1)!.

%I A229464 #20 Dec 04 2017 02:59:31
%S A229464 1,7,133,5419,383785,41782831,6472067437,1352114646163,
%T A229464 366325440650449,124893891684358615,52323557348796456661,
%U A229464 26420766706149889279867,15824833185409769038803193,11092546337733020334329204479,8995627147680234199615065312445
%N A229464 Binomial transform of (2*n + 1)!.
%C A229464 Companion sequence to A064570.
%F A229464 a(n) = Sum_{k = 0..n} binomial(n,k)*(2*k + 1)!.
%F A229464 Clearly a(n) is always odd; indeed, for n >= 1, a(n) = 1 + 6*n*b(n-1), where b(n) = [1, 11, 301, 15991, 1392761, ...] is the binomial transform of A051618.
%F A229464 a(n) = Integral_{x >= 0} x*(1 + x^2)^n*exp(-x) dx.
%F A229464 a(n) = (2*n + 1)*A064570(n) - 2*n*A064570(n-1).
%F A229464 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) = 7.
%F A229464 O.g.f.: Sum_{k >= 0} (2*k + 1)!*x^k/(1 - x)^(k + 1) = 1 + 7*x + 133*x^2 + 5419*x^3 + ....
%F A229464 a(n) ~ sqrt(Pi) * 2^(2*n + 2) * n^(2*n + 3/2) / exp(2*n). - _Vaclav Kotesovec_, Oct 30 2017
%F A229464 From _Peter Bala_, Nov 26 2017: (Start)
%F A229464 E.g.f.: exp(x)*Sum_{n >= 0} A000407(n)*x^n.
%F A229464 a(k) = a(0) (mod k) for all k (by the inhomogeneous recurrence equation).
%F A229464 More generally a(n+k) = a(n) (mod k) for all n and k (by an induction argument on n).
%F A229464 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, 7, 3, 9, 5, 1, 7, 3, 9, 5, ... with exact period 5. (End)
%e A229464 a(3) = 1*1! + 3*3! + 3*5! + 1*7! = 5419.
%t A229464 Table[Sum[Binomial[n, k] * (2*k+1)!, {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Oct 30 2017 *)
%Y A229464 Cf. A000522, A051618, A064570, A294352, A000407.
%K A229464 nonn,easy
%O A229464 0,2
%A A229464 _Peter Bala_, Sep 25 2013