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Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).

proposed

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allocatedMeandric numbers for a river crossing up to 19 parallel roads at Robertn Pricepoints.

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964691, 19302316, 54898417, 179696558

0,3

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 19 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Cf. A005316 (sequence for one road; extensive references and links).

Cf. A076876 (sequence for two parallel roads).

Cf. A206432 (sequence for unlimited number of parallel roads).

Cf. A076875, A076906, A076907.

allocated

nonn,more

Robert Price, May 07 2012

approved

editing

allocated for Robert Price

recycled

allocated

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approved

Triangle sequence, expansion of (1/(1 - x^2) + 1/(1 + x*(x + y)))

2, 0, -1, 0, 0, 1, 0, 2, 0, -1, 2, 0, -3, 0, 1, 0, -3, 0, 4, 0, -1, 0, 0, 6, 0, -5, 0, 1, 0, 4, 0, -10, 0, 6, 0, -1, 2, 0, -10, 0, 15, 0, -7, 0, 1, 0, -5, 0, 20, 0, -21, 0, 8, 0, -1, 0, 0, 15, 0, -35, 0, 28, 0, -9, 0, 1

0,1

The Row sums are repeats of:{2, -1, 1, 1, 0, 0}. The row sums behave as a D_3 Molien expansion (see Mathematica section).

The sequence is a triangle sequence from the infinite dihedral group two matrix representation by a Molien polynomial using Dixon's alpha as the negative integers to avoid alpha=2. The resulting triangle sequence resembles the Chebyshev triangle sequence A049310 and there is a possibility the individual polynomials are orthogonal.

John D.Dixon, Problems in Group Theory, Dover, 1973, p. 66, p 157, Problem 11.4, infinite dihedral group.

Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18.

Martin Burrows, Representation Theory of Finite Groups,Academic Press, New York,1965,p.111.

Larry Smith, Polynomial Invariants of Finite Groups, A.K. Peters, 1995, page125.

J-P Serre, Linear Representations of Finite Groups, Springer, 1977, p.40;

W.Fulton and J. Harris, Representation Theory, Springer, 1991, p.40.

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973,page 222.

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972, p 126.

{2},

{0, -1},

{0, 0, 1},

{0, 2, 0, -1},

{2, 0, -3, 0, 1},

{0, -3, 0, 4, 0, -1},

{0, 0, 6, 0, -5, 0, 1},

{0, 4, 0, -10, 0, 6, 0, -1},

{2, 0, -10, 0, 15, 0, -7, 0, 1},

{0, -5, 0, 20, 0, -21, 0, 8, 0, -1},

{0, 0, 15, 0, -35, 0, 28, 0, -9, 0, 1},

{0, 6, 0, -35, 0, 56, 0, -36, 0, 10, 0, -1},

{2, 0, -21, 0, 70, 0, -84, 0, 45, 0, -11, 0, 1},

{0, -7, 0, 56, 0, -126, 0, 120, 0, -55, 0, 12, 0, -1},

{0, 0, 28, 0, -126, 0, 210, 0, -165, 0, 66, 0, -13, 0, 1},

{0, 8, 0, -84, 0, 252, 0, -330, 0, 220, 0, -78, 0, 14, 0, -1}

s[1] = {{1, 0}, {-y, -1}}; s[2] = {{0, -1}, {1, -y}};

f[x_] = FullSimplify[ExpandAll[(1/2)*Sum[1/Det[IdentityMatrix[2] - x*Inverse[s[i]]], {i, 1, 2}]]]; a = Table[CoefficientList[SeriesCoefficient[Series[2*f[x], {x, 0, 15}], n], y], {n, 0, 15}]; Flatten[a]

(* Row sums as D_3 Molien sequence expansion*);

e[k_] = Exp[2*Pi*I/k]; s[1] = {{e[k], 0}, {0, 1/e[k]}}; s[2] = {{0, 1}, {1, 0}}; f[x_] = FullSimplify[ExpandAll[(1/2)*Sum[1/Det[IdentityMatrix[2] - x*s[i]], {i, 1, 2}]]] /. k -> 3; b = Table[SeriesCoefficient[Series[2*f[x], {x, 0, 100}], n], {n, 0, 100}]

A082642,A049310,A115066,A107995,A137335,A194509,A008616,

A158609.

tabl,sign,changed

recycled

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 27 2012

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Roger L. Bagula: Poisson's Kernal as a D_Infinity/ Chebyshev like group:
Clear[s, x, f, g, a, n, a0]
Solve[1/2 ((a0 c0)/((a0 - x) (c0 + x)) + 1/(1 + x^2 - 2 x y)) == (1 - 
     x^2)/(1 + x^2 - 2 x y), a0]
{{a0 -> ((c0 + x) (-1 + 2 x^2))/(-1 + 3 c0 x + 2 x^2 - 2 c0 y)}}
a0 = ((c0 + x) (-1 + 2 x^2))/(-1 + 3 c0 x + 2 x^2 - 2 c0 y)
b0 = y; c0 = 1;
s[1] = FullSimplify[ExpandAll[{{a0, 0}, {-b0, -c0}}]]
s[2] = {{0, 1}, {-1, 2*y}};
(1 - x^2)/(1 + x^2 - 2 x y)
a = Table[CoefficientList[SeriesCoefficient[
    Series[f[x], {x, 0, 15}], n], y], {n, 0, 15}]
{{1}, {0, 2}, {-2, 0, 4}, {0, -6, 0, 8}, {2, 0, -16, 0, 16}, {0, 10, 
  0, -40, 0, 32}, {-2, 0, 36, 0, -96, 0, 64}, {0, -14, 0, 112, 
  0, -224, 0, 128}, {2, 0, -64, 0, 320, 0, -512, 0, 256}, {0, 18, 
  0, -240, 0, 864, 0, -1152, 0, 512}, {-2, 0, 100, 0, -800, 0, 2240, 
  0, -2560, 0, 1024}, {0, -22, 0, 440, 0, -2464, 0, 5632, 0, -5632, 0,
   2048}, {2, 0, -144, 0, 1680, 0, -7168, 0, 13824, 0, -12288, 0, 
  4096}, {0, 26, 0, -728, 0, 5824, 0, -19968, 0, 33280, 0, -26624, 0, 
  8192}, {-2, 0, 196, 0, -3136, 0, 18816, 0, -53760, 0, 78848, 
  0, -57344, 0, 16384}, {0, -30, 0, 1120, 0, -12096, 0, 57600, 
  0, -140800, 0, 184320, 0, -122880, 0, 32768}}
Which appears to be an important result too.

Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18;.

STATUS

Roger L. Bagula: I put the references back in:
I don't see any reason why the world should suffer on account of
T.D. Noe being a horrible fellow as an editor, do you?
Since three of the references are new and are part of my self study
of Lie algebra, representation theory and group theory,
I think they should be in sequence which is a new
Chebyshev like orthogonal sequence
and probably important.
I just got the Poisson Kernal ( polynomial) as a result of a D_Infinity 
expansion as well.

Roger L. Bagula: As abusive as my treatment has been,
I still tried to do what was asked of me.