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A166919
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Coefficients of product polynomial:p(x,n) = Product[ -k - x + x^k, {k, 1, n}]
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1, -1, 2, 1, -1, -6, -5, 2, 3, 1, -1, 24, 26, -3, -14, -13, -2, 3, 3, 1, -1, -120, -154, -11, 73, 79, 47, 13, -21, -22, -9, -1, 3, 3, 1, -1, 720, 1044, 220, -427, -547, -361, -245, -41, 142, 149, 94, 30, -8, -30, -17, -8, -1, 3, 3, 1, -1, -5040, -8028, -2584, 2769
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row sums are signed factorials :
{1, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800,...}
The model of a pyramid of games matrix polynomial is the motive for these polynomials.
It appear that the diffusion velocity on the domain {x,0,1}:
v=D[p[x,n],{x,2}]/(2*D[p[x,n],x])
has a collapse point foe n>=3 past x=0.5 that gets larger as n increases.
If we look at x as a probability measure of social activity,
and the velocity as how fast the changes take place,
then large amounts of social activity can cause a pyramidal
game social structure to collapse.
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FORMULA
| p(x,n) = Product[ -k - x + x^k, {k, 1, n}];
t(n,k)=Coefficients(p(x,n))
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EXAMPLE
| {1},
{-1},
{2, 1, -1},
{-6, -5, 2, 3, 1, -1},
{24, 26, -3, -14, -13, -2, 3, 3, 1, -1},
{-120, -154, -11, 73, 79, 47, 13, -21, -22, -9, -1, 3, 3, 1, -1},
{720, 1044, 220, -427, -547, -361, -245, -41, 142, 149, 94, 30, -8, -30, -17, -8, -1, 3, 3, 1, -1},
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MATHEMATICA
| p[x_, n_] = Product[ -k - x + x^k, {k, 1, n}];
a=Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];
Flatten[a]
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CROSSREFS
| Sequence in context: A144655 A199063 A140956 * A168641 A143185 A157635
Adjacent sequences: A166916 A166917 A166918 * A166920 A166921 A166922
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KEYWORD
| sign,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Oct 23 2009
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