%I
%S 1,3,7,13,23,37,57,83,118,162,218,286,370,470,590,730,895,1086,1308,
%T 1562,1854,2186,2564,2990,3471,4010,4614,5286,6034,6862,7778,8786,
%U 9895,11110,12441,13893,15477,17199,19071,21099,23296,25669,28232,30992
%N Polynomial expansion of: p(x)=1/(1  3 x + 2 x^2 + 2 x^3  4 x^4 + 4 x^5  2 x^6  2 ^7 + 3 x^8  x^9  x^17 + 3 x^18  2 x^19  2 x^20 + 4 x^21  4 x^22 + 2 x^23 + 2 x^24  3 x^25 + x^26); polynomial associated with substitution matrix of A126810, frequency of characters in English text.
%C The substitution of a letter by the relative frequency that the letter
%C appears in English gives matrix whose polynomial has an expansion sequence
%C given here from the toral inverse.
%F a(n)=Expansion(1/(1  3 x + 2 x^2 + 2 x^3  4 x^4 + 4 x^5  2 x^6  2 ^7 + 3 x^8  x^9  x^17 + 3 x^18  2 x^19  2 x\^20 + 4 x^21  4 x^22 + 2 x^23 + 2 x^24  3 x^25 + x^26));
%t Clear[a, m, x];
%t (* English Letter Frequency*)
%t a = {3, 20, 12, 10, 1, 16, 17, 8, 5, 23, 22, 11, 14, 6, 4, 19, 25, 9, 7, 2, 13, 21, 15, 24, 18, 26};
%t (* substitution matrix*) m = Table[If[n  a[[m]] == 0, 1, 0], {n, 1, Length[a]}, {m, 1, Length[a]}];
%t p[x_] = CharacteristicPolynomial[m, x];
%t (* toral inverse*) q[x_] = 1/Expand[x^26*p[1/x]];
%t Table[SeriesCoefficient[ Series[q[t], {t, 0, 60}], n], {n, 0, 60}]
%Y Cf. A126810
%K nonn,uned
%O 0,2
%A _Roger L. Bagula_, Aug 26 2009
