

A159291


A twoway probability integer distribution function:t(n,m)=If[m <= (less than equal) Floor[n/2], a*m + b, a*(n  m) + b]*If[m <= (less than equal) Floor[m/2], a*n + b, a*(m  n) + b].


0



1, 3, 1, 5, 3, 1, 7, 9, 3, 1, 9, 15, 15, 3, 1, 11, 21, 25, 15, 3, 1, 13, 27, 35, 35, 15, 3, 1, 15, 33, 45, 49, 35, 15, 3, 1, 17, 39, 55, 63, 63, 35, 15, 3, 1, 19, 45, 65, 77, 81, 63, 35, 15, 3, 1, 21, 51, 75, 91, 99, 99, 63, 35, 15, 3, 1
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OFFSET

0,2


COMMENTS

Row sums are:
{1, 4, 3, 4, 23, 52, 101, 164, 255, 364, 509, 676, 887, 1124, 1413, 1732,
2111, 2524, 3005, 3524, 4119,...},
The first example in books that give probability distributions are "tent" integer based distributions which sum to one.
This distribution runs the tent in both the n and m directions at the same time and still gets an over all sum of one when normalized by the row sums.
Table[Sum[t[n, m]/s[n], {m, 0, n}], {n, 0, 10}]
The plots of the distributions gives skew long tail distributions.
When the negative sign is not used they have a quantumpotentiallike form, somewhat like a Morse potential.
The maximal values are squarelike: {1, 1, 3, 9, 15, 25, 35, 49, 63, 81, 99....}
This submission is by one of "The April Fool boys".


REFERENCES

E. Atlee Jackson, Equilibrium Statistical Mechanics, PrenticeHall,Inc., 1968,page 14, figure 4


LINKS

Table of n, a(n) for n=0..65.


FORMULA

t(n,m)=If[m <= (less than equal) Floor[n/2], a*m + b, a*(n  m) + b]*If[m <= (less than equal) Floor[m/2], a*n + b, a*(m  n) + b].


EXAMPLE

{1},
{3, 1},
{5, 3, 1},
{7, 9, 3, 1},
{9, 15, 15, 3, 1},
{11, 21, 25, 15, 3, 1},
{13, 27, 35, 35, 15, 3, 1},
{15, 33, 45, 49, 35, 15, 3, 1},
{17, 39, 55, 63, 63, 35, 15, 3, 1},
{19, 45, 65, 77, 81, 63, 35, 15, 3, 1},
{21, 51, 75, 91, 99, 99, 63, 35, 15, 3, 1}


MATHEMATICA

Clear[t, n, m, s, p, a, b];
a = 2; b = 1;
t[n_, m_] = If[m <= Floor[n/2], a*m + b, a*(n  m) + b]*If[m <= Floor[m/2], a*n + b, a*(m  n) + b];
s[n_] = Sum[t[n, m], {m, 0, n}];
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%];


CROSSREFS

Sequence in context: A133094 A300437 A208607 * A122510 A102662 A142048
Adjacent sequences: A159288 A159289 A159290 * A159292 A159293 A159294


KEYWORD

sign,tabl,uned


AUTHOR

Roger L. Bagula, Apr 08 2009


STATUS

approved



