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%I #8 Apr 12 2022 18:29:44
%S 1,1,1,1,-3,1,1,6,6,1,1,-3,6,-3,1,1,-21,-21,-21,-21,1,1,24,168,-84,
%T 168,24,1,1,195,-1560,5460,5460,-1560,195,1,1,-111,7215,28860,202020,
%U 28860,7215,-111,1,1,-3072,-113664,-3694080,29552640,29552640,-3694080
%N Triangle: q=3; m=2; t(n,k)=If[m == 0, n!, Product[Sum[(-1)^i*StirlingS2[ k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].
%e {1},
%e {1, 1},
%e {1, -3, 1},
%e {1, 6, 6, 1},
%e {1, -3, 6, -3, 1},
%e {1, -21, -21, -21, -21, 1},
%e {1, 24, 168, -84, 168, 24, 1},
%e {1, 195, -1560, 5460, 5460, -1560, 195, 1},
%e {1, -111, 7215, 28860, 202020, 28860, 7215, -111, 1},
%e {1, -3072, -113664, -3694080, 29552640, 29552640, -3694080, -113664, -3072, 1},
%e {1, -4053, -4150272, 76780032, -4990702080, 5703659520, -4990702080, 76780032, -4150272, -4053, 1}
%t t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^i* StirlingS2[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
%t b[n_, k_, m_] = f[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
%t Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
%K sign,tabl,uned
%O 0,5
%A _Roger L. Bagula_, Feb 10 2009
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