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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
1, 3, 7, 13, 23, 37, 57, 83, 118, 162, 218, 286, 370, 470, 590, 730, 895, 1086, 1308, 1562, 1854, 2186, 2564, 2990, 3471, 4010, 4614, 5286, 6034, 6862, 7778, 8786, 9895, 11110, 12441, 13893, 15477, 17199, 19071, 21099, 23296, 25669, 28232, 30992 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The substitution of a letter by the relative frequency that the letter

appears in English gives matrix whose polynomial has an expansion sequence

given here from the toral inverse.

LINKS

Table of n, a(n) for n=0..43.

FORMULA

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));

MATHEMATICA

Clear[a, m, x];

(* English Letter Frequency*)

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};

(* substitution matrix*) m = Table[If[n - a[[m]] == 0, 1, 0], {n, 1, Length[a]}, {m, 1, Length[a]}];

p[x_] = CharacteristicPolynomial[m, x];

(* toral inverse*) q[x_] = 1/Expand[x^26*p[1/x]];

Table[SeriesCoefficient[ Series[q[t], {t, 0, 60}], n], {n, 0, 60}]

CROSSREFS

Cf. A126810

Sequence in context: A303853 A075321 A258030 * A131205 A256309 A058682

Adjacent sequences:  A164784 A164785 A164786 * A164788 A164789 A164790

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Aug 26 2009

STATUS

approved

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Last modified August 1 14:14 EDT 2021. Contains 346391 sequences. (Running on oeis4.)