A073278 - OEIS

%I #14 Nov 21 2024 22:26:16

%S 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,

%T -1,0,21,0,-105,0,105,0,1,0,-28,0,210,0,-420,0,105,-1,0,36,0,-378,0,

%U 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

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

%C 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.

%C All coefficients are triangular numbers. The second nonzero diagonal are the triangular numbers (A000217), the third nonzero diagonal are the tritriangular numbers (A050534), etc.

%C 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

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

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

%F Sum_{k=0..n} |T(n, k)| = A000085(n). - _Peter Luschny_, Jan 10 2023

%e 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.

%e Triangle T(n, k) starts:

%e [0]  1;

%e [1] -1, 0;

%e [2]  1, 0,   -1;

%e [3] -1, 0,   3, 0;

%e [4]  1, 0,  -6, 0,    3;

%e [5] -1, 0,  10, 0,  -15, 0;

%e [6]  1, 0, -15, 0,   45, 0,  -15;

%e [7] -1, 0,  21, 0, -105, 0,  105, 0;

%e [8]  1, 0, -28, 0,  210, 0, -420, 0,  105;

%e [9] -1, 0,  36, 0, -378, 0, 1260, 0, -945, 0;

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

%Y Cf. A000085.

%K sign,tabl,changed

%O 0,9

%A _Robert G. Wilson v_, Jul 23 2002