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A144503 Sum of n-th antidiagonal of array in A144502.

%I A144503 #18 Oct 10 2023 05:05:04
%S A144503 1,2,5,20,121,982,9933,120168,1692273,27196522,491229653,9851789564,
%T A144503 217230600041,5223386190526,136025271553693,3813930989693904,
%U A144503 114553954962370785,3669540489785558994,124878930607671376549,4499311042365955114724,171098698540513965736025
%N A144503 Sum of n-th antidiagonal of array in A144502.
%H A144503 Seiichi Manyama, <a href="/A144503/b144503.txt">Table of n, a(n) for n = 0..404</a>
%F A144503 a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n - 1/2) / exp(n-1). - _Vaclav Kotesovec_, Apr 06 2019
%F A144503 a(n) = 2*(n-1)*a(n-1) + a(n-2) - 2*(n-2), with a(0) = 1, a(1) = 2. - _G. C. Greubel_, Oct 09 2023
%t A144503 a[n_]:= a[n]= If[n<2, n+1, 2*(n-1)*a[n-1] +a[n-2] -2*(n-2)];
%t A144503 Table[a[n], {n,0,30}] (* _G. C. Greubel_, Oct 09 2023 *)
%o A144503 (Ruby)
%o A144503 def A144503(n)
%o A144503   ary = []
%o A144503   a = [1]
%o A144503   (n + 1).times{|i|
%o A144503     (1..i).each{|j|
%o A144503       a[j] *= i - j
%o A144503       a[j] += a[j - 1]
%o A144503     }
%o A144503     ary << a.inject(:+)
%o A144503     a << 0
%o A144503   }
%o A144503   ary
%o A144503 end
%o A144503 p A144503(20) # _Seiichi Manyama_, Apr 06 2019
%o A144503 (Magma) [n le 2 select n else 2*(n-2)*Self(n-1) +Self(n-2) -2*(n-3): n in [1..30]]; // _G. C. Greubel_, Oct 09 2023
%o A144503 (SageMath)
%o A144503 @CachedFunction
%o A144503 def a(n): # A144503
%o A144503     if (n<2): return n+1
%o A144503     else: return 2*(n-1)*a(n-1) + a(n-2) - 2*(n-2)
%o A144503 [a(n) for n in range(31)] # _G. C. Greubel_, Oct 09 2023
%Y A144503 Cf. A144502.
%K A144503 nonn
%O A144503 0,2
%A A144503 _David Applegate_ and _N. J. A. Sloane_, Dec 13 2008