A144139 - OEIS

%I #21 Sep 19 2022 20:59:27

%S 1,5,209,1189,3905,9701,20305,37829,64769,104005,158801,232805,330049,

%T 454949,612305,807301,1045505,1332869,1675729,2080805,2555201,3106405,

%U 3742289,4471109,5301505,6242501,7303505,8494309,9825089,11306405

%N Chebyshev polynomial of the second kind U(4,n).

%H Vincenzo Librandi, <a href="/A144139/b144139.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1 + 194*x^2 + 184*x^3 + 5*x^4)/(1 - x)^5. - _Vincenzo Librandi_, May 29 2014

%F a(n) = 16*n^4-12*n^2+1 = (4*n^2-2*n-1)*(4*n^2+2*n-1). - _Vincenzo Librandi_, May 29 2014

%F From _Klaus Purath_, Sep 08 2022: (Start)

%F a(n) = A165900(2*n)*A165900(2*n+1).

%F a(n) = A057722(2*n).

%F a(n) = 4*(Sum_{i=1..n} A193250(i)) + 1 = 4*A079414(n) + 1.

%F (End)

%t lst={}; Do[AppendTo[lst, ChebyshevU[4, n]], {n, 0, 9^2}]; lst

%t CoefficientList[Series[(1 + 194 x^2 + 184 x^3 + 5 x^4)/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 29 2014 *)

%o (Magma) [16*n^4-12*n^2+1: n in [0..40]]; // _Vincenzo Librandi_, May 29 2014

%Y Cf. A057722, A079414, A165900, A193250.

%K nonn,easy

%O 0,2

%A _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008

%E Changed offset from 1 to 0 by _Vincenzo Librandi_, May 29 2014