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Triangle T(n, k) = (-1)^n*StirlingS2(n, k)*StirlingS2(n, n-k), read by rows.
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%I #10 Mar 01 2021 02:04:40

%S 1,0,0,0,1,0,0,-3,-3,0,0,6,49,6,0,0,-10,-375,-375,-10,0,0,15,2015,

%T 8100,2015,15,0,0,-21,-8820,-105350,-105350,-8820,-21,0,0,28,33782,

%U 1014300,2893401,1014300,33782,28,0,0,-36,-117810,-8004150,-54009270,-54009270,-8004150,-117810,-36,0

%N Triangle T(n, k) = (-1)^n*StirlingS2(n, k)*StirlingS2(n, n-k), read by rows.

%C Row sums are: {1, 0, 1, -6, 61, -770, 12160, -228382, 4989621, -124262532, 3475892685, ...}.

%H G. C. Greubel, <a href="/A155999/b155999.txt">Rows n = 0..100 of the triangle, flattened</a>

%F T(n, k) = (-1)^n*StirlingS2(n, k)*StirlingS2(n, n-k)

%e Triangle begins as:

%e 1;

%e 0, 0;

%e 0, 1, 0;

%e 0, -3, -3, 0;

%e 0, 6, 49, 6, 0;

%e 0, -10, -375, -375, -10, 0;

%e 0, 15, 2015, 8100, 2015, 15, 0;

%e 0, -21, -8820, -105350, -105350, -8820, -21, 0;

%e 0, 28, 33782, 1014300, 2893401, 1014300, 33782, 28, 0;

%e 0, -36, -117810, -8004150, -54009270, -54009270, -8004150, -117810, -36, 0;

%t T[n_, k_]:= (-1)^n*StirlingS2[n, k]*StirlingS2[n, n-k];

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

%o (Sage)

%o def A155999(n,k): return (-1)^n*stirling_number2(n,k)*stirling_number2(n,n-k)

%o flatten([[A155999(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 27 2021

%o (Magma)

%o A155999:= func< n,k | (-1)^n*StirlingSecond(n, k)*StirlingSecond(n, n-k) >;

%o [A155999(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 27 2021

%o (PARI) T(n, k) = (-1)^n*stirling(n, k, 2)*stirling(n, n-k, 2);

%o matrix(10, 10, n, k, n--; k--; if (n>=k, T(n,k))) \\ _Michel Marcus_, Feb 27 2021

%Y Cf. A048993.

%K tabl,sign

%O 0,8

%A _Roger L. Bagula_, Feb 01 2009

%E Edited by _G. C. Greubel_, Feb 27 2021