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A143389 Coefficient Expansion sequence of a Weaver Morse Code polynomial (using Cyclotomic prime base dot, dash, letter space and word space symbols): p(x) = -5 - 10 x - 12 x^2 - 10 x^3 - 7 x^4 - 3 x^5 + 5 x^7 + 8 x^8 + 9 x^9 + 8 x^10 + 6 x^11 + 3 x^12 + x^13. 0

%I

%S 1,-3,3,1,-6,7,-1,-9,11,7,-34,32,23,-95,99,27,-219,250,76,-571,619,

%T 241,-1517,1684,511,-3927,4500,1205,-10120,11628,3041,-26200,30648,

%U 7148,-68161,80975,16901,-176402,212169,39547,-456228,557737,91154,-1183066,1466383

%N Coefficient Expansion sequence of a Weaver Morse Code polynomial (using Cyclotomic prime base dot, dash, letter space and word space symbols): p(x) = -5 - 10 x - 12 x^2 - 10 x^3 - 7 x^4 - 3 x^5 + 5 x^7 + 8 x^8 + 9 x^9 + 8 x^10 + 6 x^11 + 3 x^12 + x^13.

%D Claude Shannon and Warren Weaver, A Mathematical Theory of Communication, University of Illinois Press, Chicago, 1963, pp. 37-38.

%F p(x) = -5 - 10 x - 12 x^2 - 10 x^3 - 7 x^4 - 3 x^5 + 5 x^7 + 8 x^8 + 9 x^9 + 8 x^10 + 6 x^11 + 3 x^12 + x^13; a(n) = Coefficient_expansion(x^13*p(1/x)).

%F G.f.: -1/(5*x^13+10*x^12+12*x^11+10*x^10+7*x^9+3*x^8-5*x^6-8*x^5 -9*x^4 -8*x^3-6*x^2-3*x-1).

%e Weaver determinant:

%e A0 = Cyclotomic[2, x]

%e B0 = Cyclotomic[5, x]

%e C0 = Cyclotomic[3, x]

%e D0 = Cyclotomic[7, x]

%e Expand[FullSimplify[ExpandAll[((1 + x) (1 + x + x^2) (

%e 1 + x + x^2 + x^3 + x^4) (

%e 1 + x + x^2 + x^3 + x^4 + x^5 + x^6))*Det[{{-1, (1/B0 + 1/A0)}, {(1/

%e D0 + 1/C0),

%e 1/A0 + 1/B0 - 1}}]]]]

%t p[x_] = -5 - 10 x - 12 x^2 - 10 x^3 - 7 x^4 - 3 x^5 + 5 x^7 + 8 x^8 + 9 x^9 + 8 x^10 + 6 x^11 + 3 x^12 + x^13; q[x_] = ExpandAll[x^13*p[1/x]]; a = Table[SeriesCoefficient[Series[1/q[x], {x, 0, 30}], n], {n, 0, 30}]

%K uned,sign

%O 0,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 22 2008

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Last modified May 14 19:53 EDT 2021. Contains 343903 sequences. (Running on oeis4.)