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A155999
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Triangle T(n, k) = (-1)^n*StirlingS2(n, k)*StirlingS2(n, n-k), read by rows.
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1
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1, 0, 0, 0, 1, 0, 0, -3, -3, 0, 0, 6, 49, 6, 0, 0, -10, -375, -375, -10, 0, 0, 15, 2015, 8100, 2015, 15, 0, 0, -21, -8820, -105350, -105350, -8820, -21, 0, 0, 28, 33782, 1014300, 2893401, 1014300, 33782, 28, 0, 0, -36, -117810, -8004150, -54009270, -54009270, -8004150, -117810, -36, 0
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OFFSET
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0,8
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COMMENTS
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Row sums are: {1, 0, 1, -6, 61, -770, 12160, -228382, 4989621, -124262532, 3475892685, ...}.
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LINKS
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FORMULA
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T(n, k) = (-1)^n*StirlingS2(n, k)*StirlingS2(n, n-k)
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EXAMPLE
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Triangle begins as:
1;
0, 0;
0, 1, 0;
0, -3, -3, 0;
0, 6, 49, 6, 0;
0, -10, -375, -375, -10, 0;
0, 15, 2015, 8100, 2015, 15, 0;
0, -21, -8820, -105350, -105350, -8820, -21, 0;
0, 28, 33782, 1014300, 2893401, 1014300, 33782, 28, 0;
0, -36, -117810, -8004150, -54009270, -54009270, -8004150, -117810, -36, 0;
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MATHEMATICA
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T[n_, k_]:= (-1)^n*StirlingS2[n, k]*StirlingS2[n, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
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PROG
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(Sage)
def A155999(n, k): return (-1)^n*stirling_number2(n, k)*stirling_number2(n, n-k)
(Magma)
A155999:= func< n, k | (-1)^n*StirlingSecond(n, k)*StirlingSecond(n, n-k) >;
(PARI) T(n, k) = (-1)^n*stirling(n, k, 2)*stirling(n, n-k, 2);
matrix(10, 10, n, k, n--; k--; if (n>=k, T(n, k))) \\ Michel Marcus, Feb 27 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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