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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

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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.

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

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) 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