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A triangular sequence of the antidiagonal of the Cyclotomic q like factorial: t(n,m)=If[m == 0, n!, Product[Cyclotomic[k, m + 1], {k, 1, n}]].
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%I #4 Dec 09 2016 06:14:33

%S 1,1,1,1,1,2,1,2,3,6,1,3,8,21,24,1,4,15,104,105,120,1,5,24,315,1040,

%T 3255,720,1,6,35,744,5355,125840,9765,5040,1,7,48,1505,19344,1826055,

%U 880880,1240155,40320,1,8,63,2736,55685,15107664,23738715,962801840

%N A triangular sequence of the antidiagonal of the Cyclotomic q like factorial: t(n,m)=If[m == 0, n!, Product[Cyclotomic[k, m + 1], {k, 1, n}]].

%C Row sums are:

%C {1, 2, 4, 12, 57, 349, 5360, 146786, 4008315, 1023152227, 210533527258,...}.

%C The Cyclotomic binomials turn out rational.

%F t(n,m)=If[m == 0, n!, Product[Cyclotomic[k, m + 1], {k, 1, n}]]; out_(n,m)=antidiagonal(t(n,m))

%e {1},

%e {1, 1},

%e {1, 1, 2},

%e {1, 2, 3, 6},

%e {1, 3, 8, 21, 24},

%e {1, 4, 15, 104, 105, 120},

%e {1, 5, 24, 315, 1040, 3255, 720},

%e {1, 6, 35, 744, 5355, 125840, 9765, 5040},

%e {1, 7, 48, 1505, 19344, 1826055, 880880, 1240155, 40320},

%e {1, 8, 63, 2736, 55685, 15107664, 23738715, 962801840, 21082635, 362880},

%e {1, 9, 80, 4599, 136800, 86590175, 317260944, 129637122615, 78949750880, 1539032355, 3628800}

%t Clear[t, n, m, i, k, a, b];

%t t[n_, m_] = If[m == 0, n!, Product[Cyclotomic[k, m + 1], {k, 1, n}]];

%t a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];

%t b = Table[Table[a[[m, n - m + 1]], {m, n, 1, -1}], {n, 1, Length[a]}];

%t Flatten[%]

%K nonn,tabl

%O 0,6

%A _Roger L. Bagula_ and _Gary W. Adamson_, Feb 10 2009