A112742 a(n) = n^2*(n^2 - 1)/3.

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

Offset: 0

Author

Matthew T. Cornick (maruth(AT)gmail.com), Sep 16 2005

Keywords

• nonn
• easy

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

Examples

			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.
		

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).

Crossrefs

Cf. A000914, A002415, A002492.

Programs

Mathematica

Table[D[ChebyshevT[n, x], {x, 2}], {n, 0, 100}] /. x -> 1

PARI

a(n)=n^2*(n^2-1)/3 \\ Charles R Greathouse IV, Oct 07 2015

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

a(n) = A053126(n+2) - A006324(n-1). - Yasser Arath Chavez Reyes, Feb 22 2024