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 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]. 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified June 15 22:06 EDT 2021. Contains 345053 sequences. (Running on oeis4.)