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A122101 T(n,k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A000670(n-k+i).

%I A122101 #21 Jun 23 2023 07:28:55
%S A122101 1,1,0,3,2,2,13,10,8,6,75,62,52,44,38,541,466,404,352,308,270,4683,
%T A122101 4142,3676,3272,2920,2612,2342,47293,42610,38468,34792,31520,28600,
%U A122101 25988,23646,545835,498542,455932,417464,382672,351152,322552,296564,272918
%N A122101 T(n,k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A000670(n-k+i).
%H A122101 G. C. Greubel, <a href="/A122101/b122101.txt">Rows n = 0..100 of triangle, flattened</a>
%F A122101 Doubly-exponential generating function: Sum_{n, k} a(n-k,k) x^n/n! y^k/k! = exp(-y)/(2-exp(x+y)).
%e A122101 Triangle begins as:
%e A122101      1;
%e A122101      1,    0;
%e A122101      3,    2,    2;
%e A122101     13,   10,    8,    6;
%e A122101     75,   62,   52,   44,   38;
%e A122101    541,  466,  404,  352,  308,  270;
%e A122101   4683, 4142, 3676, 3272, 2920, 2612, 2342;
%e A122101   ...
%p A122101 T:= (n, k)-> k!*(n-k)!*coeff(series(coeff(series(exp(-y)/
%p A122101         (2-exp(x+y)), y, k+1), y, k), x, n-k+1), x, n-k):
%p A122101 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Oct 02 2019
%p A122101 # second Maple program:
%p A122101 b:= proc(n) option remember; `if`(n<2, 1,
%p A122101       add(b(n-j)*binomial(n, j), j=1..n))
%p A122101     end:
%p A122101 T:= (n, k)-> add(binomial(k, j)*(-1)^j*b(n-j), j=0..k):
%p A122101 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Oct 02 2019
%t A122101 A000670[n_]:= If[n==0,1,Sum[k! StirlingS2[n, k], {k, n}]]; T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[k, j]*A000670[n-k+j], {j,0,k}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 02 2019 *)
%o A122101 (PARI)
%o A122101 A000670(n) = sum(k=0,n, k!*stirling(n,k,2));
%o A122101 T(n,k) = sum(j=0,k, (-1)^(k-j)*binomial(k, j)*A000670(n-k+j));
%o A122101 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 02 2019
%o A122101 (Magma)
%o A122101 A000670:= func< n | &+[Factorial(k)*StirlingSecond(n,k): k in [0..n]] >;
%o A122101 [(&+[(-1)^(k-j)*Binomial(k,j)*A000670(n-k+j): j in [0..k]]): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Oct 02 2019
%o A122101 (Sage)
%o A122101 def A000670(n): return sum(factorial(k)*stirling_number2(n,k) for k in (0..n))
%o A122101 def T(n,k): return sum((-1)^(k-j)*binomial(k, j)*A000670(n-k+j) for j in (0..k))
%o A122101 [[T(n,k) for k in (0..n)] for n in (0..10)]
%o A122101 (GAP)
%o A122101 A000670:= function(n)
%o A122101      return Sum([0..n], i-> Factorial(i)*Stirling2(n,i) ); end;
%o A122101 T:= function(n,k)
%o A122101     return Sum([0..k], j-> (-1)^(k-j)*Binomial(k, j)*A000670(n-k+j) ); end;
%o A122101 Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Oct 02 2019
%Y A122101 Columns k=0-1 give: A000670, A232472.
%Y A122101 Row sums give A089677(n+1).
%Y A122101 Main diagonal gives A052841.
%Y A122101 T(2n,n) gives A340837.
%Y A122101 Cf. A005649, A069321, A073146.
%K A122101 easy,nonn,tabl
%O A122101 0,4
%A A122101 _Vladeta Jovovic_, Oct 18 2006