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A156588 A triangle of q factorial type based on Stirling first polynomials: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

%I A156588 #5 Dec 09 2016 06:16:07
%S A156588 1,1,1,1,-1,2,1,-1,2,6,1,-1,3,-12,24,1,-1,4,-36,288,120,1,-1,5,-80,
%T A156588 2160,-34560,720,1,-1,6,-150,9600,-777600,24883200,5040,1,-1,7,-252,
%U A156588 31500,-8064000,1959552000,-125411328000,40320,1,-1,8,-392,84672,-52920000
%N A156588 A triangle of q factorial type based on Stirling first polynomials: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].
%C A156588 Row sums are:
%C A156588 {1, 2, 2, 8, 15, 376, -31755, 24120096, -123459768425, 5017134314247168,
%C A156588 -1827769039991244222327,...}.
%F A156588 t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
%F A156588 out_(n,k)=Antidiagonal(t(n,k)).
%e A156588 {1},
%e A156588 {1, 1},
%e A156588 {1, -1, 2},
%e A156588 {1, -1, 2, 6},
%e A156588 {1, -1, 3, -12, 24},
%e A156588 {1, -1, 4, -36, 288, 120},
%e A156588 {1, -1, 5, -80, 2160, -34560, 720},
%e A156588 {1, -1, 6, -150, 9600, -777600, 24883200, 5040},
%e A156588 {1, -1, 7, -252, 31500, -8064000, 1959552000, -125411328000, 40320},
%e A156588 {1, -1, 8, -392, 84672, -52920000, 54190080000, -39504568320000, 5056584744960000, 362880},
%e A156588 {1, -1, 9, -576, 197568, -256048128, 800150400000, -3277416038400000, 7167708875980800000, -1834933472251084800000, 3628800}
%t A156588 Clear[t, n, m, i, k, a, b];
%t A156588 t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
%t A156588 a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
%t A156588 b = Table[Table[a[[m, n - m + 1]], {m, n, 1, -1}], {n, 1, Length[a]}];
%t A156588 Flatten[%]
%Y A156588 A009963
%K A156588 sign,tabl,uned
%O A156588 0,6
%A A156588 _Roger L. Bagula_, Feb 10 2009