

A112742


a(n) = n^2*(n^21)/3.


6



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
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OFFSET

0,3


COMMENTS

Second derivative of the nth 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, qp and qp.  Wesley Ivan Hurt, Apr 15 2018


LINKS

Table of n, a(n) for n=0..36.
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) = (n1)*n^2*(n+1)/3 = 4*A002415(n).
a(n) = 2*( A000914(n1) + C(n+1,4) )  David Scambler, Nov 27 2006
a(n) = 5*a(n1)10*a(n2)+10*a(n3)5*a(n4)+a(n5). G.f.: 4*x^2*(1+x)/(1x)^5.  Colin Barker, Jan 26 2012


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^21)/3 \\ Charles R Greathouse IV, Oct 07 2015


CROSSREFS

Cf. A000914, A002415, A002492.
Sequence in context: A199904 A250132 A025220 * A158494 A209456 A069145
Adjacent sequences: A112739 A112740 A112741 * A112743 A112744 A112745


KEYWORD

nonn,easy


AUTHOR

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


STATUS

approved



