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Search: id:A112742
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| A112742 |
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Second derivative of the n-th Chebyshev polynomial (of the first kind) evaluated at x=1. |
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+0
1
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| 0, 0, 4, 24, 80, 200, 420, 784, 1344, 2160, 3300, 4840, 6864, 9464, 12740, 16800, 21760, 27744, 34884, 43320, 53200, 64680, 77924, 93104, 110400, 130000, 152100, 176904, 204624, 235480, 269700, 307520, 349184, 394944, 445060, 499800, 559440
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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The second derivative at x=-1 is just (-1)^n * a(n)
The difference between two consecutive terms, n+1 and n, generates the sequence b(n)=a(n+1)-a(n) which is A002492.
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LINKS
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Chebyshev polynomials of the first kind
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FORMULA
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a(n) = (n-1)*n*n*(n+1)/3;
a(n) = 2*( A000914(n-1) + C(n+1,4) ) - David J. Scambler (dscambler(AT)bmm.com), Nov 27 2006
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EXAMPLE
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a(4)=80 because:
C_4(x) = 1 - 8x^2 + 8x^4
C'_4(x) = -16x+32x^3
C''_4(x) = -16+96x^2
C''_4(1) = -16+96 = 80
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MATHEMATICA
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Table[D[ChebyshevT[n, x], {x, 2}], {n, 0, 100}] /. x -> 1
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CROSSREFS
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Adjacent sequences: A112739 A112740 A112741 this_sequence A112743 A112744 A112745
Sequence in context: A112611 A011915 A025220 this_sequence A069145 A005561 A061612
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KEYWORD
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nonn
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AUTHOR
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Matthew T. Cornick (maruth(AT)gmail.com), Sep 16 2005
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