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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
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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
| Chebyshev polynomials of the first kind
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FORMULA
| 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
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). G.f.: 4*x^2*(1+x)/(1-x)^5. [Colin Barker, Jan 26 2012]
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EXAMPLE
| 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
| Table[D[ChebyshevT[n, x], {x, 2}], {n, 0, 100}] /. x -> 1
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CROSSREFS
| Sequence in context: A011915 A199904 A025220 * A158494 A069145 A005561
Adjacent sequences: A112739 A112740 A112741 * A112743 A112744 A112745
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KEYWORD
| nonn
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AUTHOR
| Matthew T. Cornick (maruth(AT)gmail.com), Sep 16 2005
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