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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)).

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