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A157895
Coefficients of polynomials of a prime like factor set : p(x,n)=Sum[x^i, {i, 0, (Prime[n] - 1)/2}]; q(n,n)=Sum[(-1)^i*x^i, {i, 0, (Prime[n] - 1)/2}]; t(x,n)=If[n == 0, 1, If[n == 1, x + 1, (x + 1)*p[x, n]*q[x, n]]].
0
1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
0,1
COMMENTS
Row sums are:
{1, 2, 0, 6, 0, 0, 14, 18, 0, 0, 30, 0, 38, 42, 0, 0, 54, 0, 62, 0, 0,...}.
This row sum minus one picks out as cyclotomic the primes; A002144:
{5,13,17,29,37,41,53,61,...}
FORMULA
p(x,n)=Sum[x^i, {i, 0, (Prime[n] - 1)/2}];
q(n,n)=Sum[(-1)^i*x^i, {i, 0, (Prime[n] - 1)/2}];
t(x,n)=If[n == 0, 1, If[n == 1, x + 1, (x + 1)*p[x, n]*q[x, n]]];
out_(n,m)=coefficients(t(x,n)).
EXAMPLE
{1},
{1, 1},
{1, 1, -1, -1},
{1, 1, 1, 1, 1, 1},
{1, 1, 1, 1, -1, -1, -1, -1},
{1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
MATHEMATICA
Clear[p, q, t, x, n];
p[x_, n_] := Sum[x^i, {i, 0, (Prime[n] - 1)/2}];
q[x_, n_] := Sum[(-1)^i*x^i, {i, 0, (Prime[n] - 1)/2}];
t[x_, n_] := If[n == 0, 1, If[n == 1, x + 1, (x + 1)*p[x, n]*q[x, n]]];
Table[ExpandAll[t[x, n]], {n, 0, 10}];
Table[CoefficientList[ExpandAll[t[x, n]], x], {n, 0, 10}];
Flatten[%]
CROSSREFS
Sequence in context: A232544 A309873 A162511 * A077008 A158387 A265643
KEYWORD
sign,tabl,uned
AUTHOR
Roger L. Bagula, Mar 08 2009
STATUS
approved