OFFSET
0,3
COMMENTS
Second derivative of the n-th Chebyshev polynomial (of the first kind) evaluated at x=1.
The second derivative at x=-1 is just (-1)^n * a(n).
The difference between two consecutive terms generates the sequence a(n+1) - a(n) = A002492(n).
Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, |q-p| and |q-p|. - Wesley Ivan Hurt, Apr 15 2018
LINKS
Eric Weisstein's World of Mathematics, Chebyshev polynomials of the first kind
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = (n-1)*n^2*(n+1)/3 = 4*A002415(n).
a(n) = 2*( A000914(n-1) + C(n+1,4) ). - David Scambler, Nov 27 2006
From Colin Barker, Jan 26 2012: (Start)
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. (End)
E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/3. - Stefano Spezia, Dec 11 2021
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.
MATHEMATICA
Table[D[ChebyshevT[n, x], {x, 2}], {n, 0, 100}] /. x -> 1
PROG
(PARI) a(n)=n^2*(n^2-1)/3 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Matthew T. Cornick (maruth(AT)gmail.com), Sep 16 2005
STATUS
approved