%I #21 Sep 08 2022 08:45:52

%S 1,1,1,1,-1,-1,1,-8,-2,-2,1,-23,43,19,19,1,-49,301,-199,-79,-79,1,-89,

%T 1186,-3314,796,76,76,1,-146,3529,-22196,34644,-2400,2640,2640,1,-223,

%U 8793,-100967,372863,-362529,3375,-36945,-36945,1,-323,19333,-361691,2466883,-6010901,3911515,-33509,329371,329371

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

%H G. C. Greubel, <a href="/A176153/b176153.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n, k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n, j).

%F T(n, n) = A317274(n). - _G. C. Greubel_, Aug 03 2021

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, -1, -1;

%e 1, -8, -2, -2;

%e 1, -23, 43, 19, 19;

%e 1, -49, 301, -199, -79, -79;

%e 1, -89, 1186, -3314, 796, 76, 76;

%e 1, -146, 3529, -22196, 34644, -2400, 2640, 2640;

%e 1, -223, 8793, -100967, 372863, -362529, 3375, -36945, -36945;

%p seq(seq( add(combinat[stirling1](n,n-j)*binomial(n,j), j=0..k), k=0..n), n=0..10); # _G. C. Greubel_, Nov 26 2019

%t T[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j,0,k}];

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

%o (PARI) T(n,k) = sum(j=0,k, stirling(n,n-j,1)*binomial(n,j)); \\ _G. C. Greubel_, Nov 26 2019

%o (Magma) [(&+[StirlingFirst(n, n-j)*Binomial(n,j): j in [0..k]]): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 26 2019

%o (Sage) [[sum((-1)^j*stirling_number1(n, n-j)*binomial(n,j) for j in (0..k)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 26 2019

%o (GAP) Flat(List([0..10], n-> List([0..n], k-> Sum([0..k], j-> (-1)^j*Stirling1(n,n-j)* Binomial(n,j)) ))); # _G. C. Greubel_, Nov 26 2019

%Y Cf. A008277, A176154, A176155, A176156, A176157, A317274.

%K sign,tabl

%O 0,8

%A _Roger L. Bagula_, Apr 10 2010