A112742 - OEIS

%I #35 Mar 22 2024 17:48:18

%S 0,0,4,24,80,200,420,784,1344,2160,3300,4840,6864,9464,12740,16800,

%T 21760,27744,34884,43320,53200,64680,77924,93104,110400,130000,152100,

%U 176904,204624,235480,269700,307520,349184,394944,445060,499800,559440

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

%C Second derivative of the n-th Chebyshev polynomial (of the first kind) evaluated at x=1.

%C The second derivative at x=-1 is just (-1)^n * a(n).

%C The difference between two consecutive terms generates the sequence a(n+1) - a(n) = A002492(n).

%C 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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">Chebyshev polynomials of the first kind</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = (n-1)*n^2*(n+1)/3 = 4*A002415(n).

%F a(n) = 2*( A000914(n-1) + C(n+1,4) ). - _David Scambler_, Nov 27 2006

%F From _Colin Barker_, Jan 26 2012: (Start)

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

%F G.f.: 4*x^2*(1+x)/(1-x)^5. (End)

%F E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/3. - _Stefano Spezia_, Dec 11 2021

%F a(n) = A053126(n+2) - A006324(n-1). - _Yasser Arath Chavez Reyes_, Feb 22 2024

%e a(4)=80 because

%e C_4(x) = 1 - 8x^2 + 8x^4,

%e C'_4(x) = -16x + 32x^3,

%e C''_4(x) = -16 + 96x^2,

%e C''_4(1) = -16 + 96 = 80.

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

%o (PARI) a(n)=n^2*(n^2-1)/3 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A000914, A002415, A002492.

%K nonn,easy

%O 0,3

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