

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
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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



