%I #13 Jun 09 2018 06:50:36
%S 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,
%T -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,
%U -45,0,210,0,-210,0,45,0,-1
%N 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)).
%C Polynomials like these are seen in complex dynamics.
%C 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)).
%C From _Johannes W. Meijer_, Mar 10 2012: (Start)
%C The row sums equal A146559 and the two antidiagonal sums lead to A104862 (minus a(0)) and A110161 (minus a(0)).
%C The mirror of this triangle (for the absolute values of the coefficients) is A119467. (End)
%H G. C. Greubel, <a href="/A141665/b141665.txt">Rows n=0..100 of triangle, flattened</a>
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Kimberling/kimberling56.html">Polynomials associated with reciprocation</a>, JIS 12 (2009) 09.3.4, section 5.
%F p(x,n) = (1+I*x)^n
%F t(n,m) = real part of coefficients(p(x,n))
%F s(n,m) = imaginary part of coefficients(p(x,n))
%e s(n,m) = imaginary part of coefficients(p(x,n))
%e {0},
%e {0, 1},
%e {0, 2, 0},
%e {0, 3, 0, -1},
%e {0, 4, 0, -4, 0},
%e {0, 5, 0, -10, 0, 1},
%e {0, 6, 0, -20, 0, 6, 0},
%e {0, 7, 0, -35, 0, 21, 0, -1},
%e {0, 8, 0, -56, 0, 56, 0, -8, 0},
%e {0, 9, 0, -84, 0, 126, 0, -36, 0, 1},
%e {0, 10, 0, -120, 0, 252, 0, -120, 0, 10, 0}
%p From _Johannes W. Meijer_, Mar 10 2012: (Start)
%p 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);
%p 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)
%t 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[%]
%Y Cf. A007318, A146559, A104862, A110161, A119467.
%K easy,sign,tabl
%O 0,9
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 05 2008
%E Edited and information added by _Johannes W. Meijer_, Mar 10 2012