OFFSET
0,9
COMMENTS
Polynomials like these are seen in complex dynamics.
This method symmetrically breaks up Pascal's triangle A007318 into two parts as polynomial coefficient vectors. See the examples for the s(n,m) = imaginary part of coefficients(p(x,n)).
From Johannes W. Meijer, Mar 10 2012: (Start)
The row sums equal A146559 and the two antidiagonal sums lead to A104862 (minus a(0)) and A110161 (minus a(0)).
The mirror of this triangle (for the absolute values of the coefficients) is A119467. (End)
LINKS
G. C. Greubel, Rows n=0..100 of triangle, flattened
Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4, section 5.
FORMULA
p(x,n) = (1+I*x)^n
t(n,m) = real part of coefficients(p(x,n))
s(n,m) = imaginary part of coefficients(p(x,n))
EXAMPLE
s(n,m) = imaginary part of coefficients(p(x,n))
{0},
{0, 1},
{0, 2, 0},
{0, 3, 0, -1},
{0, 4, 0, -4, 0},
{0, 5, 0, -10, 0, 1},
{0, 6, 0, -20, 0, 6, 0},
{0, 7, 0, -35, 0, 21, 0, -1},
{0, 8, 0, -56, 0, 56, 0, -8, 0},
{0, 9, 0, -84, 0, 126, 0, -36, 0, 1},
{0, 10, 0, -120, 0, 252, 0, -120, 0, 10, 0}
MAPLE
From Johannes W. Meijer, Mar 10 2012: (Start)
nmax:=10: for n from 0 to nmax do p(x, n) := (1+I*x)^n: for m from 0 to n do t(n, m) := Re(coeff(p(x, n), x, m)) od: od: seq(seq(t(n, m), m=0..n), n=0..nmax);
nmax:=10: for n from 0 to nmax do for m from 0 to n do A119467(n, m) := binomial(n, m) * (1+(-1)^(n-m))/2: if (m mod 4 = 2) then x(n, m):= -1 else x(n, m):= 1 end if: od: od: for n from 0 to nmax do for m from 0 to n do t(n, m) := A119467(n, n-m)*x(n, m) od: od: seq(seq(t(n, m), m=0..n), n=0..nmax); # (End)
MATHEMATICA
p[x_, n_] := If[n == 0, 1, Product[(1 + I*x), {i, 1, n}]]; Table[Expand[p[x, n]], {n, 0, 10}]; Table[Im[CoefficientList[p[x, n], x]], {n, 0, 10}]; Flatten[%] Table[Re[CoefficientList[p[x, n], x]], {n, 0, 10}]; Flatten[%]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula and Gary W. Adamson, Sep 05 2008
EXTENSIONS
Edited and information added by Johannes W. Meijer, Mar 10 2012
STATUS
approved