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Search: id:A136667
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| A136667 |
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Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Hermite polynomial of the Hochstadt kind (A137286)as related to the generalized Chebyshev in a Shabat way(A123583): p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p)x,n)^2. |
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| 0, 1, 0, -1, -3, 0, 4, 0, -1, 1, 0, -25, 0, 10, 0, -1, -63, 0, 144, 0, -97, 0, 18, 0, -1, 1, 0, -1089, 0, 924, 0, -262, 0, 28, 0, -1, -2303, 0, 8352, 0, -9489, 0, 3576, 0, -574, 0, 40, 0, -1, 1, 0, -77841, 0, 103230, 0, -49291, 0, 10548, 0, -1099, 0, 54, 0, -1, -147455, 0, 748800, 0, -1215585, 0, 699630, 0, -188043, 0
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row sums are:
{0, 0, 0, -15, 1, -399, -399, -14399, -78399, -639999, -12959999};
From conversion experiments with generalized Hermite to Chebyshev types,
I conjectured a theorem (unproved):
If there exists a generalized Shabat/ Chebyshev polynomial set,
there is a corresponding
by linear transform Hermite polynomial set:
Such that the domains and weight functions are conformally connected.
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REFERENCES
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Defined :page 8 and pages 42 - 43: Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
G. B. Shabat and I. A. Voevodskii, Drawing curves over number fields, The Grothendieck Festschift, vol. 3, Birkhaeuser, 1990, pp. 199-22
G. B. Shabat and A. Zvonkin, Plane trees and algebraic numbers, Contemporary Math., 1994, vol. 178, pp. 233-275.
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FORMULA
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out=1-A137286(x,n)^2; p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p)x,n)^2
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EXAMPLE
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{0},
{1, 0, -1},
{-3, 0, 4, 0, -1},
{1, 0, -25, 0, 10, 0, -1},
{-63, 0, 144, 0, -97, 0, 18, 0, -1},
{1, 0, -1089, 0, 924, 0, -262,0, 28, 0, -1},
{-2303, 0, 8352, 0, -9489, 0, 3576, 0, -574, 0, 40, 0, -1},
{1, 0, -77841, 0, 103230, 0, -49291, 0, 10548,0, -1099, 0, 54, 0, -1},
{-147455, 0, 748800, 0, -1215585, 0, 699630, 0, -188043, 0, 26100, 0, -1915, 0, 70, 0, -1},
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MATHEMATICA
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P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; Q[x_, n_] := Q[x, n] = 1 - P[x, n]^2; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a = Join[{{0}}, Table[CoefficientList[Q[x, n], x], {n, 0, 10}]]; Flatten[a]
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CROSSREFS
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Cf. A123583, A137286, A136247.
Sequence in context: A128252 A033596 A063529 this_sequence A004588 A066705 A027636
Adjacent sequences: A136664 A136665 A136666 this_sequence A136668 A136669 A136670
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 02 2008
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