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A073278 A triangle constructed from the coefficients of the n-th derivative of the normal probability distribution function. 1
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; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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.

REFERENCES

Samuel M. Selby, Editor-in-Chief, CRC Standard Mathematical Tables, 21st Edition, 1973, pp. 582.

LINKS

Table of n, a(n) for n=0..77.

FORMULA

a(n) = the coefficient list of the x's of the n-th d(e^(-x^2 /2)/dx.

EXAMPLE

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; ...

MATHEMATICA

y = E^(-x^2/2); Flatten[ Table[ Reverse[ CoefficientList[ Dt[y, {x, n}]/y, x]], {n, 0, 11} ]]

CROSSREFS

Sequence in context: A056100 A141665 A136689 * A081658 A187253 A022904

Adjacent sequences:  A073275 A073276 A073277 * A073279 A073280 A073281

KEYWORD

sign

AUTHOR

Robert G. Wilson v, Jul 23 2002

STATUS

approved

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Last modified January 19 00:40 EST 2020. Contains 331030 sequences. (Running on oeis4.)