A159291 A two-way 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].

-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

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 quantum-potential-like form, somewhat like a Morse potential.

The maximal values are square-like: {-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, Prentice-Hall,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 A350279

Adjacent sequences: A159288 A159289 A159290 * A159292 A159293 A159294

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Apr 08 2009

Status

approved