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.
REFERENCES
Samuel M. Selby, Editor-in-Chief, CRC Standard Mathematical Tables, 21st Edition, 1973, pp. 582.
FORMULA
a(n) is the coefficient list of the x's of the n-th d(e^(-x^2 /2)/dx.
Sum_{k=0..n} |T(n, k)| = A000085(n). - Peter Luschny, Jan 10 2023
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.
Triangle T(n, k) starts:
[0] 1;
[1] -1, 0;
[2] 1, 0, -1;
[3] -1, 0, 3, 0;
[4] 1, 0, -6, 0, 3;
[5] -1, 0, 10, 0, -15, 0;
[6] 1, 0, -15, 0, 45, 0, -15;
[7] -1, 0, 21, 0, -105, 0, 105, 0;
[8] 1, 0, -28, 0, 210, 0, -420, 0, 105;
[9] -1, 0, 36, 0, -378, 0, 1260, 0, -945, 0;
MATHEMATICA
y = E^(-x^2/2); Flatten[ Table[ Reverse[ CoefficientList[ Dt[y, {x, n}]/y, x]], {n, 0, 11} ]]
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Robert G. Wilson v, Jul 23 2002
STATUS
approved