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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 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.)