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A111885 Row sums of triangle A112492.

%I A111885 #9 Jul 25 2023 09:15:05
%S A111885 1,2,5,20,152,2542,100326,10194844,2809233510,2212797607312,
%T A111885 5359196565766782,39928779843430949176,1018129474625651322506886,
%U A111885 85890171235256453902613870992,26477529277143069417959927152215342
%N A111885 Row sums of triangle A112492.
%H A111885 G. C. Greubel, <a href="/A111885/b111885.txt">Table of n, a(n) for n = 0..55</a>
%F A111885 a(n) = Sum_{j=0..n} A112492(n, j), n >= 0.
%t A111885 T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, (k+1)^(n-k)*T[n-1, k-1] + k!*T[n-1, k]];
%t A111885 a[n_]:= a[n]= Sum[T[n,k], {k,0,n}]; (* T = A112492 *)
%t A111885 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Jul 24 2023 *)
%o A111885 (Magma)
%o A111885 T:= func< n,k | (-1)*Factorial(k+1)^(n-k)*(&+[(-1)^j*Binomial(k+1,j)/j^(n-k) : j in [1..k+1]]) >; // T = A112492
%o A111885 A111885:= func< n | (&+[T(n,k): k in [0..n]]) >;
%o A111885 [A111885(n): n in [0..40]]; // _G. C. Greubel_, Jul 24 2023
%o A111885 (SageMath)
%o A111885 @CachedFunction
%o A111885 def T(n,k): # T = A112492
%o A111885     if (k==0 or k==n): return 1
%o A111885     else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
%o A111885 def A111885(n): return sum(T(n,k) for k in range(n+1))
%o A111885 [A111885(n) for n in range(31)] # _G. C. Greubel_, Jul 24 2023
%Y A111885 Cf. A112492.
%K A111885 nonn,easy
%O A111885 0,2
%A A111885 _Wolfdieter Lang_, Sep 12 2005