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A166919 Coefficients of product polynomial:p(x,n) = Product[ -k - x + x^k, {k, 1, n}] 0
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; text; internal format)
OFFSET

0,3

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 for 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.

LINKS

Table of n, a(n) for n=0..60.

FORMULA

p(x,n) = Product[ -k - x + x^k, {k, 1, n}];

t(n,k)=Coefficients(p(x,n))

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},

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]

CROSSREFS

Sequence in context: A330490 A199063 A140956 * A338874 A338876 A260238

Adjacent sequences:  A166916 A166917 A166918 * A166920 A166921 A166922

KEYWORD

sign,uned

AUTHOR

Roger L. Bagula, Oct 23 2009

STATUS

approved

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Last modified July 25 19:34 EDT 2021. Contains 346291 sequences. (Running on oeis4.)