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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)). 1
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A100574 A342122 A056100 * A136689 A359364 A073278
KEYWORD
easy,sign,tabl
AUTHOR
EXTENSIONS
Edited and information added by Johannes W. Meijer, Mar 10 2012
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)