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Triangle T(n, k) = n!*StirlingS2(n, k)/binomial(n, k), read by rows.
1

%I #7 Sep 08 2022 08:45:41

%S 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 3240,3120,1800,720,0,720,15120,43344,50400,33600,15120,5040,0,5040,

%U 182880,695520,979776,756000,383040,141120,40320,0,40320,2570400,13068000,22377600,20018880,11430720,4656960,1451520,362880

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

%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 99.

%H G. C. Greubel, <a href="/A156815/b156815.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = n!*StirlingS2(n, k)/binomial(n, k).

%F From _G. C. Greubel_, Jun 10 2021: (Start)

%F T(n, 1) = T(n, n) = n!.

%F T(n, 2) = 2*A029767(n+1).

%F T(n, n-1) = A180119(n). (End)

%e Triangle begins as:

%e 1;

%e 0, 1;

%e 0, 1, 2;

%e 0, 2, 6, 6;

%e 0, 6, 28, 36, 24;

%e 0, 24, 180, 300, 240, 120;

%e 0, 120, 1488, 3240, 3120, 1800, 720;

%e 0, 720, 15120, 43344, 50400, 33600, 15120, 5040;

%e 0, 5040, 182880, 695520, 979776, 756000, 383040, 141120, 40320;

%t T[n_, k_] = n!*StirlingS2[n, k]/Binomial[n, k];

%t Table[T[n, k], {n, 0, 12}, {k,0,n}]//Flatten

%o (Magma) [Factorial(n)*StirlingSecond(n,k)/Binomial(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 10 2021

%o (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 Cf. A048993, A029767, A180119.

%K nonn,tabl

%O 0,6

%A _Roger L. Bagula_, Feb 16 2009

%E Edited by _G. C. Greubel_, Jun 10 2021