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Search: id:A133156
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| A133156 |
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Irregular triangle read by rows: coefficients of U(n,x), Chebyshev polynomials of the second kind with exponents in decreasing order. |
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+0
2
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| 1, 2, 4, -1, 8, -4, 16, -12, 1, 32, -32, 6, 64, -80, 24, -1, 128, -192, 80, -8, 256, -448, 240, -40, 1, 512, -1024, 672, -160, 10
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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The Chebyshev polynomials of the second kind are defined by the recurrence relation: U(0,x) = 1; U(1,x) = 2x; U(n+1,x) = 2x*U(n,x) - U(n-1,x).
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REFERENCES
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Wikipedia, (Chebyshev polynomials).
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FORMULA
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One example of a generating function for U(n) = 1/(1 - 2tx + t^2). Given A038207, shift down columns to allow for (1, 1, 2, 2, 3, 3,...) terms in each row, then insert alternate signs.
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EXAMPLE
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The first few Chebyshev polynomials of the second kind are:
1;
2x;
4x^2 - 1;
8x^3 - 4x;
16x^4 - 12x^2 + 1;
32x^5 - 32x^3 + 6x;
64x^6 - 80x^4 + 24x^2 - 1;
128x^7 - 192x^5 + 80x^3 - 8x;
256x^8 - 448x^6 + 240x^4 - 40x^2 + 1;
512x^9 - 1024x^7 _ 672x^5 - 160x^3 + 10x;
...
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CROSSREFS
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Cf. A038207, A053117.
Sequence in context: A112931 A121685 A125810 this_sequence A127529 A091977 A112829
Adjacent sequences: A133153 A133154 A133155 this_sequence A133157 A133158 A133159
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KEYWORD
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nonn,tabf
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 16 2007
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