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.

1, -3, 3, 1, -6, 7, -1, -9, 11, 7, -34, 32, 23, -95, 99, 27, -219, 250, 76, -571, 619, 241, -1517, 1684, 511, -3927, 4500, 1205, -10120, 11628, 3041, -26200, 30648, 7148, -68161, 80975, 16901, -176402, 212169, 39547, -456228, 557737, 91154, -1183066, 1466383

Offset

0,2

References

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

Links

Table of n, a(n) for n=0..44.

Formula

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)).

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).

Example

Weaver determinant:
A0 = Cyclotomic[2, x]
B0 = Cyclotomic[5, x]
C0 = Cyclotomic[3, x]
D0 = Cyclotomic[7, x]
Expand[FullSimplify[ExpandAll[((1 + x) (1 + x + x^2) (
1 + x + x^2 + x^3 + x^4) (
1 + x + x^2 + x^3 + x^4 + x^5 + x^6))*Det[{{-1, (1/B0 + 1/A0)}, {(1/
D0 + 1/C0),
1/A0 + 1/B0 - 1}}]]]]

Mathematica

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}]

Crossrefs

Sequence in context: A010468 A082009 A110640 * A219218 A208524 A094040

Adjacent sequences: A143386 A143387 A143388 * A143390 A143391 A143392

Keyword

uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Oct 22 2008

Status

approved