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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

G. C. Greubel, Rows n=0..100 of triangle, flattened

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

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

Sequence in context: A046665 A100574 A056100 * A136689 A073278 A081658

Adjacent sequences:  A141662 A141663 A141664 * A141666 A141667 A141668

KEYWORD

easy,sign,tabl

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified September 23 03:04 EDT 2020. Contains 337291 sequences. (Running on oeis4.)