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Triangle read by rows based on the Stirling numbers S1: t(n,m)=Sum[(-1)^(n + 1)* StirlingS1[n, j]*(k + 1 - j)^(n - 1), {j, 0, k + 1}].
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%I #9 May 03 2015 11:53:51

%S 1,2,5,6,37,80,24,334,1179,2644,120,3566,20617,63413,146394,720,44316,

%T 413608,1766365,5161687,12157088,5040,632052,9362908,55669771,

%U 207499100,590541383,1411732608,40320,10212336,236604140,1953603356,9326112285

%N Triangle read by rows based on the Stirling numbers S1: t(n,m)=Sum[(-1)^(n + 1)* StirlingS1[n, j]*(k + 1 - j)^(n - 1), {j, 0, k + 1}].

%C Row sums are: {1, 7, 123, 4181, 234110, 19543784, 2275442862, 352293774104, 69988577590464,...}.

%C The sum algorithm is based on the Eulerian number sum with Stirling first kind substituted for the binomial.

%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%F t(n,m)=Sum[(-1)^(n + 1)* StirlingS1[n, j]*(k + 1 - j)^(n - 1), {j, 0, k + 1}].

%e {1},

%e {2, 5},

%e {6, 37, 80},

%e {24, 334, 1179, 2644},

%e {120, 3566, 20617, 63413, 146394},

%e {720, 44316, 413608, 1766365, 5161687, 12157088},

%e {5040, 632052, 9362908, 55669771, 207499100, 590541383, 1411732608},

%e {40320, 10212336, 236604140, 1953603356, 9326112285, 32221533668, 90256527071, 218289140928},

%e {362880, 184767984, 6618132828, 75520418032, 462351260321, 1945272980967, 6403986114493, 17752922644079, 43341720908880}

%t Clear[t, n, k]; t[n_, k_] = Sum[(-1)^(n + 1)* StirlingS1[n, j]*(k + 1 - j)^(n - 1), {j, 0, k + 1}];

%t Table[Table[t[n, k], {k, 1, n - 1}], {n, 2, 10}];

%t Flatten[%]

%K nonn,tabl

%O 2,2

%A _Roger L. Bagula_, Dec 15 2008