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A164787 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. 0

%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

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Last modified September 26 20:34 EDT 2021. Contains 347672 sequences. (Running on oeis4.)