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Antidiagonal triangle sequence of coefficient expansion of the general prime product polynomial: f(x,n) = (1 + t^2)/Product_{i=1..n} (1 - t^prime(i + 1)).
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%I #4 Dec 09 2016 06:10:50

%S 1,1,0,1,0,1,1,0,1,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,1,0,2,1,1,0,1,1,0,2,

%T 1,0,1,0,1,1,0,2,1,1,1,1,0,1,1,0,2,1,2,2,1,1,0,1,1,0,2,1,2,2,1,0,1,0,

%U 1,1,0,2,1,2,2,2,2,1,1,0,1,1,0,2,1,2,2,2,3,2,1,1,0,1,1,0,2,1,2,2,2,3,2,2,0

%N Antidiagonal triangle sequence of coefficient expansion of the general prime product polynomial: f(x,n) = (1 + t^2)/Product_{i=1..n} (1 - t^prime(i + 1)).

%C Row sums are {1, 1, 2, 3, 3, 4, 6, 6, 8, 11, 11, 15, 18, 19, 25}.

%F f(x,n) = (1 + t^2)/Product_{i=1..n} (1 - t^prime(i + 1)); t(n,m) = expansion(f(x,n)); out_n,m(antidiagonal) = t(n-m+1,n).

%e {1},

%e {1, 0},

%e {1, 0, 1},

%e {1, 0, 1, 1},

%e {1, 0, 1, 1, 0},

%e {1, 0, 1, 1, 0, 1},

%e {1, 0, 1, 1, 0, 2, 1},

%e {1, 0, 1, 1, 0, 2, 1, 0},

%e {1, 0, 1, 1, 0, 2, 1, 1, 1},

%e {1, 0, 1, 1, 0, 2, 1, 2, 2, 1},

%e {1, 0, 1, 1, 0, 2, 1, 2, 2, 1, 0},

%e {1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 2, 1},

%e {1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 2, 1},

%e {1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 2, 2, 0},

%e {1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 3, 4, 2, 1}

%t Clear[f, b, a] f[t_, n_] := (1 + t^2)/Product[1 - t^Prime[i + 1], {i, 1, n}]; a = Table[Table[SeriesCoefficient[Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {m, 1, 31}]; b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}] ; Flatten[b]

%K nonn,uned

%O 1,27

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 22 2008