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A159291
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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].
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0
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-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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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".
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REFERENCES
| E. Atlee Jackson, Equilibrium Statistical Mechanics, Prentice-Hall,Inc., 1968,page 14, figure 4
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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].
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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}
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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[%];
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CROSSREFS
| Sequence in context: A130301 A133601 A133094 * A122510 A102662 A142048
Adjacent sequences: A159288 A159289 A159290 * A159292 A159293 A159294
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KEYWORD
| sign,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 08 2009
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