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%I #8 Apr 03 2021 03:03:47
%S 1,-1,2,4,-6,6,-36,44,-36,24,576,-600,420,-240,120,-14400,13152,-8100,
%T 4080,-1800,720,518400,-423360,233856,-105840,42000,-15120,5040,
%U -25401600,18817920,-9455040,3898944,-1411200,463680,-141120,40320,1625702400,-1104606720,510295680,-193777920,64653120,-19595520,5503680,-1451520,362880
%N Triangle T(n, k) = n! * StirlingS1(n, k)/binomial(n, k), read by rows.
%C Row sums are: {1, 1, 4, -4, 276, -6348, 254976, -13188096, 887086080, ...}.
%H G. C. Greubel, <a href="/A142473/b142473.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n, k) = n! * StirlingS1(n, k)/ binomial(n, k).
%e The triangle begins as:
%e 1;
%e -1, 2;
%e 4, -6, 6;
%e -36, 44, -36, 24;
%e 576, -600, 420, -240, 120;
%e -14400, 13152, -8100, 4080, -1800, 720;
%e 518400, -423360, 233856, -105840, 42000, -15120, 5040;
%e -25401600, 18817920, -9455040, 3898944, -1411200, 463680, -141120, 40320;
%p A142473:= (n,k)-> n!*Stirling1(n,k)/binomial(n,k);
%p seq(seq(A142473(n,k), k=1..n), n=1..12); # _G. C. Greubel_, Apr 02 2021
%t T[n_, k_]:= n!*StirlingS1[n, k]/Binomial[n, k];
%t Table[T[n, k], {n,12},{k,n}]//Flatten (* modified by _G. C. Greubel_, Apr 02 2021 *)
%o (Magma)
%o A142473:= func< n,k | Factorial(n)*StirlingFirst(n,k)/Binomial(n,k) >;
%o [A142473(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 02 2021
%o (Sage)
%o def A142473(n,k): return (-1)^(n-k)*factorial(n)*stirling_number1(n, k)/binomial(n,k)
%o flatten([[A142473(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 02 2021
%Y Cf. A008275.
%K sign,tabl
%O 1,3
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 21 2008
%E Edited by _G. C. Greubel_, Apr 02 2021
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