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A176153 Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j), read by rows.

%I A176153 #21 Sep 08 2022 08:45:52
%S A176153 1,1,1,1,-1,-1,1,-8,-2,-2,1,-23,43,19,19,1,-49,301,-199,-79,-79,1,-89,
%T A176153 1186,-3314,796,76,76,1,-146,3529,-22196,34644,-2400,2640,2640,1,-223,
%U A176153 8793,-100967,372863,-362529,3375,-36945,-36945,1,-323,19333,-361691,2466883,-6010901,3911515,-33509,329371,329371
%N A176153 Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j), read by rows.
%H A176153 G. C. Greubel, <a href="/A176153/b176153.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176153 T(n, k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n, j).
%F A176153 T(n, n) = A317274(n). - _G. C. Greubel_, Aug 03 2021
%e A176153 Triangle begins as:
%e A176153   1;
%e A176153   1,    1;
%e A176153   1,   -1,   -1;
%e A176153   1,   -8,   -2,      -2;
%e A176153   1,  -23,   43,      19,     19;
%e A176153   1,  -49,  301,    -199,    -79,     -79;
%e A176153   1,  -89, 1186,   -3314,    796,      76,   76;
%e A176153   1, -146, 3529,  -22196,  34644,   -2400, 2640,   2640;
%e A176153   1, -223, 8793, -100967, 372863, -362529, 3375, -36945, -36945;
%p A176153 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 A176153 T[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j,0,k}];
%t A176153 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o A176153 (PARI) T(n,k) = sum(j=0,k, stirling(n,n-j,1)*binomial(n,j)); \\ _G. C. Greubel_, Nov 26 2019
%o A176153 (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 A176153 (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 A176153 (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 A176153 Cf. A008277, A176154, A176155, A176156, A176157, A317274.
%K A176153 sign,tabl
%O A176153 0,8
%A A176153 _Roger L. Bagula_, Apr 10 2010