A152575 A triangle of coefficients of polynomials with roots as the Pi-digits base ten A000796(n)=d(n):d(1)=3; p(x,n)=-d(1)*Product[x-d(m),{m,2,n}].

-3, 3, -3, -12, 15, -3, 12, -27, 18, -3, -60, 147, -117, 33, -3, 540, -1383, 1200, -414, 60, -3, -1080, 3306, -3783, 2028, -534, 66, -3, 6480, -20916, 26004, -15951, 5232, -930, 84, -3, -32400, 111060, -150936, 105759, -42111, 9882, -1350, 99, -3, 97200

Offset

1,1

Comments

p(x,n)=Product[(x-i)^a(i),{i,0,9}]: a(i) is the count number of the

way the digits occur by the n-th digit.

The limiting polynomial is:

pL(x)=Product[(x-i),{i,0,9}]=

-362880 x + 1026576 x^2 - 1172700 x^3 + 723680 x^4 - 269325 x^5 + 63273 x^6 - 9450 x^7 + 870 x^8 - 45 x^9 + x^10;

since if the digits occur equally:

p(x,Infinity)=-3*Product[(x-i),{i,0,9}]^(Infinity/10).

Or at the n-th digits equality:

p(x,n)=-3*Product[(x-i),{i,0,9}]^(n/10).

The n+1 digit:

p(x,n+1)=p(x,n)*(x-d(n+1)).

Links

Table of n, a(n) for n=1..46.

Formula

Pi-digits base ten A000796(n)=d(n):

p(x,n)=-d(1)*Product[x-d(m),{m,2,n}].

Example

{-3},
{3, -3},
{-12,15, -3},
{12, -27, 18, -3},
{-60, 147, -117, 33, -3},
{540, -1383, 1200, -414, 60, -3},
{-1080, 3306, -3783, 2028, -534, 66, -3},
{6480, -20916, 26004, -15951, 5232, -930, 84, -3},
{-32400, 111060, -150936, 105759, -42111, 9882, -1350, 99, -3},
{97200, -365580, 563868, -468213, 232092, -71757, 13932, -1647, 108, -3}

Mathematica

Clear[a, p, n, m];
a = Delete[Flatten[RealDigits[Pi, 10, 100]], 100];
p[x_, n_] := If[n == 1, -a[[1]], -a[[1]]*Product[x - a[[m]], {m, 2, n}]];
Table[CoefficientList[p[x, n], x], {n, 1, 10}]
Flatten[%]

Crossrefs

A000796

Sequence in context: A156715 A133797 A225073 * A170851 A052900 A024947

Adjacent sequences: A152572 A152573 A152574 * A152576 A152577 A152578

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Dec 08 2008

Status

approved