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Search: id:A073278
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| A073278 |
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A triangle constructed from the coefficients of the n-th derivative of the normal probability distribution function. |
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2
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| 1, -1, 0, 1, 0, -1, -1, 0, 3, 0, 1, 0, -6, 0, 3, -1, 0, 10, 0, -15, 0, 1, 0, -15, 0, 45, 0, -15, -1, 0, 21, 0, -105, 0, 105, 0, 1, 0, -28, 0, 210, 0, -420, 0, 105, -1, 0, 36, 0, -378, 0, 1260, 0, -945, 0, 1, 0, -45, 0, 630, 0, -3150, 0, 4725, 0, -945, -1, 0, 55, 0, -990, 0, 6930, 0, -17325, 0, 10395, 0
(list; graph; listen)
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OFFSET
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0,9
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COMMENT
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The n-th derivative of the normal probability distribution function will be a polynomial of n degrees times f(x) of which every other term is zero.
All coefficients are triangular numbers. The second nonzero diagonal are the triangular numbers (A000217), the third nonzero diagonal are the tritriangular numbers (A050534), etc.
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REFERENCES
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Samuel M. Selby, Editor-in-Chief, CRC Standard Mathematical Tables, 21st Edition, 1973, pp. 582.
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FORMULA
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a(n) = the coefficient list of the x's of the n-th d(e^(-x^2 /2)/dx.
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EXAMPLE
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f(x) = 1/Sqrt(2*Pi) * e^(-x^2 /2). The polynomial involved in the seventh derivative of the f(x)/dx is (x^7 + 21x^5 - 105x^3 + 105x). Therefore the seventh antidiagonal reads the coefficients as 1, 0, 21, 0, -105, 0, 105.
1; 1,0; 1,0,1; 1,0,3,0; ...
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MATHEMATICA
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y = E^(-x^2/2); Flatten[ Table[ Reverse[ CoefficientList[ Dt[y, {x, n}]/y, x]], {n, 0, 11} ]]
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CROSSREFS
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Sequence in context: A056100 A141665 A136689 this_sequence A135481 A128311 A132884
Adjacent sequences: A073275 A073276 A073277 this_sequence A073279 A073280 A073281
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KEYWORD
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sign
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AUTHOR
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Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 23 2002
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