A106202
Expansion of Im(x/(1 - x - 2*i*x^2)), i=sqrt(-1).
Original entry on oeis.org
0, 0, 0, 2, 4, 6, 8, 2, -20, -66, -144, -230, -236, 22, 856, 2610, 5308, 7918, 7104, -4150, -36636, -100794, -193368, -269342, -198772, 274974, 1522192, 3846778, 6966452, 8986230, 4917240, -14538862, -61860772, -145127602, -248063392, -292843734, -90180988, 692992166, 2468418888, 5415220546, 8722746156, 9258303102
Offset: 0
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Table[-Im[Det[Array[KroneckerDelta[#1 + 1, #2]*I &, {n - 1, n - 1}] + Array[KroneckerDelta[#1 - 1, #2]*2 &, {n - 1, n - 1}] + IdentityMatrix[n - 1]]], {n, 2, 40}] (* John M. Campbell, Jun 04 2011 *)
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concat(vector(3), Vec(2*x^3/(1-2*x+x^2+4*x^4) + O(x^50))) \\ Michel Marcus, Jan 03 2016
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)).
Original entry on oeis.org
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
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}
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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)
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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[%]
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