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A156815 Triangle T(n, k) = n!*StirlingS2(n, k)/binomial(n, k), read by rows.

%I A156815 #7 Sep 08 2022 08:45:41
%S A156815 1,0,1,0,1,2,0,2,6,6,0,6,28,36,24,0,24,180,300,240,120,0,120,1488,
%T A156815 3240,3120,1800,720,0,720,15120,43344,50400,33600,15120,5040,0,5040,
%U A156815 182880,695520,979776,756000,383040,141120,40320,0,40320,2570400,13068000,22377600,20018880,11430720,4656960,1451520,362880
%N A156815 Triangle T(n, k) = n!*StirlingS2(n, k)/binomial(n, k), read by rows.
%D A156815 Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 99.
%H A156815 G. C. Greubel, <a href="/A156815/b156815.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A156815 T(n, k) = n!*StirlingS2(n, k)/binomial(n, k).
%F A156815 From _G. C. Greubel_, Jun 10 2021: (Start)
%F A156815 T(n, 1) = T(n, n) = n!.
%F A156815 T(n, 2) = 2*A029767(n+1).
%F A156815 T(n, n-1) = A180119(n). (End)
%e A156815 Triangle begins as:
%e A156815   1;
%e A156815   0,    1;
%e A156815   0,    1,      2;
%e A156815   0,    2,      6,      6;
%e A156815   0,    6,     28,     36,     24;
%e A156815   0,   24,    180,    300,    240,    120;
%e A156815   0,  120,   1488,   3240,   3120,   1800,    720;
%e A156815   0,  720,  15120,  43344,  50400,  33600,  15120,   5040;
%e A156815   0, 5040, 182880, 695520, 979776, 756000, 383040, 141120, 40320;
%t A156815 T[n_, k_] = n!*StirlingS2[n, k]/Binomial[n, k];
%t A156815 Table[T[n, k], {n, 0, 12}, {k,0,n}]//Flatten
%o A156815 (Magma) [Factorial(n)*StirlingSecond(n,k)/Binomial(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 10 2021
%o A156815 (Sage) flatten([[factorial(n)*stirling_number2(n,k)/binomial(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 10 2021
%Y A156815 Cf. A048993, A029767, A180119.
%K A156815 nonn,tabl
%O A156815 0,6
%A A156815 _Roger L. Bagula_, Feb 16 2009
%E A156815 Edited by _G. C. Greubel_, Jun 10 2021