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A094422 Antidiagonal sums of array A094416.

%I A094422 #10 Jan 12 2024 01:14:52
%S A094422 1,5,26,174,1531,17275,243092,4165260,85133685,2039546785,56447550542,
%T A094422 1783865468186,63766726231791,2558290237404919,114418196763735112,
%U A094422 5670168958036693976,309630356618418661737,18536683645526372648445
%N A094422 Antidiagonal sums of array A094416.
%H A094422 G. C. Greubel, <a href="/A094422/b094422.txt">Table of n, a(n) for n = 1..310</a>
%F A094422 a(n) = Sum_{k=1..n} Bo(n-k+1, k), where Bo(r, n) = A094416(r, n).
%t A094422 Table[Sum[Sum[j!*(n - k + 1)^j*StirlingS2[k, j], {j, 0, n}], {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Jul 23 2018 *)
%o A094422 (Magma)
%o A094422 A094422:= func< n | (&+[(&+[Factorial(j)*(n-k+1)^j*StirlingSecond(k,j): j in [0..n]]): k in [1..n]]) >;
%o A094422 [A094422(n): n in [1..25]]; // _G. C. Greubel_, Jan 11 2024
%o A094422 (SageMath)
%o A094422 def f(n,k,j): return factorial(j)*(n-k+1)^j*stirling_number2(k,j)
%o A094422 def A094422(n): return sum(sum(f(n,k,j) for j in range(n+1)) for k in range(1,n+1))
%o A094422 [A094422(n) for n in range(1,26)] # _G. C. Greubel_, Jan 11 2024
%Y A094422 Cf. A094416.
%K A094422 nonn
%O A094422 1,2
%A A094422 _Ralf Stephan_, May 02 2004