A141665 A signed half of Pascal's triangle A007318: p(x,n) = (1+I*x)^n; t(n,m) = real part of coefficients(p(x,n)).

1, 1, 0, 1, 0, -1, 1, 0, -3, 0, 1, 0, -6, 0, 1, 1, 0, -10, 0, 5, 0, 1, 0, -15, 0, 15, 0, -1, 1, 0, -21, 0, 35, 0, -7, 0, 1, 0, -28, 0, 70, 0, -28, 0, 1, 1, 0, -36, 0, 126, 0, -84, 0, 9, 0, 1, 0, -45, 0, 210, 0, -210, 0, 45, 0, -1

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 05 2008

Keywords

• easy
• sign
• tabl

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)

Examples

			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}
		

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.

Crossrefs

Cf. A007318, A146559, A104862, A110161, A119467.

Programs

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[%]

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))

Extensions

Edited and information added by Johannes W. Meijer, Mar 10 2012