A142598 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)).

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, 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, 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

Offset

1,27

Comments

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

Links

Table of n, a(n) for n=1..105.

Formula

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).

Example

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

Mathematica

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]

Crossrefs

Sequence in context: A263657 A261769 A005590 * A274372 A037800 A144411

Adjacent sequences: A142595 A142596 A142597 * A142599 A142600 A142601

Keyword

nonn,uned

Author

Roger L. Bagula and Gary W. Adamson, Sep 22 2008

Status

approved