|
%I #3 Mar 30 2012 17:34:28
%S -3,3,-3,-12,15,-3,12,-27,18,-3,-60,147,-117,33,-3,540,-1383,1200,
%T -414,60,-3,-1080,3306,-3783,2028,-534,66,-3,6480,-20916,26004,-15951,
%U 5232,-930,84,-3,-32400,111060,-150936,105759,-42111,9882,-1350,99,-3,97200
%N 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}].
%C p(x,n)=Product[(x-i)^a(i),{i,0,9}]: a(i) is the count number of the
%C way the digits occur by the n-th digit.
%C The limiting polynomial is:
%C pL(x)=Product[(x-i),{i,0,9}]=
%C -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;
%C since if the digits occur equally:
%C p(x,Infinity)=-3*Product[(x-i),{i,0,9}]^(Infinity/10).
%C Or at the n-th digits equality:
%C p(x,n)=-3*Product[(x-i),{i,0,9}]^(n/10).
%C The n+1 digit:
%C p(x,n+1)=p(x,n)*(x-d(n+1)).
%F Pi-digits base ten A000796(n)=d(n):
%F p(x,n)=-d(1)*Product[x-d(m),{m,2,n}].
%e {-3},
%e {3, -3},
%e {-12,15, -3},
%e {12, -27, 18, -3},
%e {-60, 147, -117, 33, -3},
%e {540, -1383, 1200, -414, 60, -3},
%e {-1080, 3306, -3783, 2028, -534, 66, -3},
%e {6480, -20916, 26004, -15951, 5232, -930, 84, -3},
%e {-32400, 111060, -150936, 105759, -42111, 9882, -1350, 99, -3},
%e {97200, -365580, 563868, -468213, 232092, -71757, 13932, -1647, 108, -3}
%t Clear[a, p, n, m];
%t a = Delete[Flatten[RealDigits[Pi, 10, 100]], 100];
%t p[x_, n_] := If[n == 1, -a[[1]], -a[[1]]*Product[x - a[[m]], {m, 2, n}]];
%t Table[CoefficientList[p[x, n], x], {n, 1, 10}]
%t Flatten[%]
%Y A000796
%K sign,tabl,uned
%O 1,1
%A _Roger L. Bagula_, Dec 08 2008
|
