A073278 Triangle read by rows. The triangle is constructed from the coefficients of the n-th derivative of the normal probability distribution function.

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

Offset: 0

Author

Robert G. Wilson v, Jul 23 2002

Keywords

• sign
• tabl

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.

If r(n,x) denotes the polynomial of integer coefficients for row n, then r(n+1,x) = diff(r(n,x), x) - x*r(n, x) is the polynomial for row n+1. This gives an effective method of computing the sequence without recourse to the exp function. - Sean A. Irvine, Nov 21 2024

Examples

			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;
		

References

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

Crossrefs

Cf. A000085.

Programs

Mathematica

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

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