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A144495 Row 2 of array in A144502.

%I A144495 #36 Oct 08 2023 04:43:56
%S A144495 2,7,30,155,946,6687,53822,486355,4877250,53759351,646098622,
%T A144495 8409146187,117836551730,1768850337295,28318532194206,481652022466307,
%U A144495 8673291031865602,164849403644999655,3297954931572397790,69274457019123638011,1524368720086682440242
%N A144495 Row 2 of array in A144502.
%H A144495 Robert Israel, <a href="/A144495/b144495.txt">Table of n, a(n) for n = 0..447</a>
%F A144495 E.g.f.: exp(x)*(2-x)/(1-x)^3.
%F A144495 a(n) = (1/2) * (floor((n+1)*(n+1)!*e) + floor(n*n!*e)). [_Gary Detlefs_, Jun 06 2010]
%F A144495 a(n) = (1/2) * ( A001339(n) + A001339(n+1) ). [_Gary Detlefs_, Jun 06 2010]
%F A144495 a(n) = (1/2) * (3 + n + (1 + 3*n + n^2) * A000522(n)). - _Gerry Martens_, Oct 02 2016
%F A144495 a(n) = ((4+3*n)*a(n-1) - (n+3)*(n-1)*a(n-2) + (n-1)*(n-2)*a(n-3))/2. - _Robert Israel_, Oct 02 2016
%F A144495 From _Peter Bala_, May 27 2022: (Start)
%F A144495 a(n) = (1/2)*(A000522(n+2) - A000522(n)).
%F A144495 a(n) = (1/2)*Sum_{k = 0..n} binomial(n,k)*(k+4)*(k+1)!; binomial transform of A038720(n+1).
%F A144495 a(n) = (1/2)*e*Integral_{x >= 1} x^n*(x^2 - 1)*exp(-x).
%F A144495 a(2*n) is even and a(2*n+1) is odd. More generally, a(n+k) == a(n) (mod k) for all n and k. It follows that for each positive integer k, the sequence obtained by reducing a(n) modulo k is periodic, with the exact period dividing k. Various divisibility properties of the sequence follow from this; for example, a(3*n+2) == 0 (mod 3), a(5*n+2) == a(5*n+3) (mod 5), a(7*n+1) == 0 (mod 7) and a(11*n+4) == 0 (mod 11). (End)
%F A144495 a(n) = (n*(n^2 + 3*n + 1)*a(n-1) - (n + 2))/(n^2 + n - 1), with a(0) = 2. - _G. C. Greubel_, Oct 07 2023
%p A144495 f:= rectoproc({a(n)=((4+3*n)*a(n-1)-(n+3)*(n-1)*a(n-2)+(n-1)*(n-2)*a(n-3))/2,a(0)=2,a(1)=7,a(2)=30},a(n),remember):
%p A144495 map(f, [$0..40]); # _Robert Israel_, Oct 02 2016
%t A144495 (* First program *)
%t A144495 t[0, _] = 1; t[n_, 0] := t[n, 0] = t[n-1, 1];
%t A144495 t[n_, k_] := t[n, k] = t[n-1, k+1] + k*t[n, k-1];
%t A144495 a[n_] := t[2, n];
%t A144495 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Aug 19 2022 *)
%t A144495 (* Second program *)
%t A144495 a[n_]:= a[n]= If[n==0, 2, (n*(n^2+3*n+1)*a[n-1] -(n+2))/(n^2+n-1)];
%t A144495 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Oct 07 2023 *)
%o A144495 (Magma)
%o A144495 A144495:= func< n | (&+[Binomial(n,k)*(k+4)*Factorial(k+1) : k in [0..n]])/2 >;
%o A144495 [A144495(n): n in [0..40]]; // _G. C. Greubel_, Oct 07 2023
%o A144495 (SageMath)
%o A144495 def A144495(n): return sum(binomial(n,j)*factorial(j+1)*(j+4) for j in range(n+1))/2
%o A144495 [A144495(n) for n in range(41)] # _G. C. Greubel_, Oct 07 2023
%Y A144495 Cf. A000522, A001339, A038720, A144496, A144497, A144498.
%Y A144495 Cf. A144499, A144500, A144501, A144502, A144503.
%K A144495 nonn
%O A144495 0,1
%A A144495 _David Applegate_ and _N. J. A. Sloane_, Dec 13 2008