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A176154 Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} Stirling1(n, n-j)*binomial(n,j), read by rows.

%I A176154 #19 Sep 08 2022 08:45:52
%S A176154 2,2,2,0,-2,0,-1,-10,-10,-1,20,-4,86,-4,20,-78,-128,102,102,-128,-78,
%T A176154 77,-13,1982,-6628,1982,-13,77,2641,2494,1129,12448,12448,1129,2494,
%U A176154 2641,-36944,-37168,12168,-463496,745726,-463496,12168,-37168,-36944
%N A176154 Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} Stirling1(n, n-j)*binomial(n,j), read by rows.
%H A176154 G. C. Greubel, <a href="/A176154/b176154.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176154 T(n, k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} Stirling1(n, n-j)*binomial(n,j).
%e A176154 Triangle begins as:
%e A176154      2;
%e A176154      2,    2;
%e A176154      0,   -2,    0;
%e A176154     -1,  -10,  -10,    -1;
%e A176154     20,   -4,   86,    -4,    20;
%e A176154    -78, -128,  102,   102,  -128,  -78;
%e A176154     77,  -13, 1982, -6628,  1982,  -13,   77;
%e A176154   2641, 2494, 1129, 12448, 12448, 1129, 2494, 2641;
%p A176154 with(combinat);
%p A176154 T:= proc(n, k) option remember; add(stirling1(n, n-j)*binomial(n, j), j=0..k) + add(stirling1(n, n-j)* binomial(n, j), j=0..n-k); end;
%p A176154 seq(seq(T(n,k), k=0..n), n=0..10); # _G. C. Greubel_, Nov 26 2019
%t A176154 T[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j, 0, k}] + Sum[StirlingS1[n, n-j]*Binomial[n, j], {j, 0, n-k}];
%t A176154 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o A176154 (PARI) T(n,k) = sum(j=0,k, stirling(n,n-j,1)*binomial(n,j)) + sum(j=0,n-k, stirling(n, n-j,1)*binomial(n,j)); \\ _G. C. Greubel_, Nov 26 2019
%o A176154 (Magma)
%o A176154 T:= func< n,k | (&+[StirlingFirst(n,n-j)*Binomial(n,j): j in [0..k]]) + (&+[StirlingFirst(n,n-j)*Binomial(n,j): j in [0..n-k]]) >;
%o A176154 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 26 2019
%o A176154 (Sage)
%o A176154 def T(n, k): return sum((-1)^j*stirling_number1(n,n-j)*binomial(n,j) for j in (0..k)) + sum((-1)^j*stirling_number1(n, n-j)*binomial(n,j) for j in (0..n-k))
%o A176154 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 26 2019
%o A176154 (GAP)
%o A176154 T:= function(n,k) return Sum([0..k], j-> (-1)^j*Stirling1(n,n-j)*Binomial(n,j)) + Sum([0..n-k], j-> (-1)^j*Stirling1(n, n-j)*Binomial(n,j)); end;
%o A176154 Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # _G. C. Greubel_, Nov 26 2019
%Y A176154 Cf. A048994, A132393, A176153, A176155, A176156, A176157.
%K A176154 sign,tabl
%O A176154 0,1
%A A176154 _Roger L. Bagula_, Apr 10 2010
%E A176154 Name edited by _G. C. Greubel_, Nov 27 2019