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A144124 P_4(2n+1), the Legendre polynomial of order 4 at 2n+1.

%I A144124 #17 Feb 16 2025 08:33:08
%S A144124 1,321,2641,10321,28401,63601,124321,220641,364321,568801,849201,
%T A144124 1222321,1706641,2322321,3091201,4036801,5184321,6560641,8194321,
%U A144124 10115601,12356401,14950321,17932641,21340321,25212001,29588001
%N A144124 P_4(2n+1), the Legendre polynomial of order 4 at 2n+1.
%C A144124 Legendre polynomial LP_4(x) = (35*x^4 - 30*x^2 + 3)/8. - _Klaus Brockhaus_, Nov 21 2009
%H A144124 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial.</a>
%H A144124 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A144124 From _Klaus Brockhaus_, Nov 21 2009: (Start)
%F A144124 a(n) = 70*n^4 + 140*n^3 + 90*n^2 + 20*n + 1.
%F A144124 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 1680 for n > 3; a(0)=1, a(1)=321, a(2)=2641, a(3)=10321.
%F A144124 G.f.: (1 + 316*x + 1046*x^2 + 316*x^3 + x^4)/(1-x)^5. (End)
%t A144124 Table[LegendreP[4,2n+1],{n,0,50}] (* _N. J. A. Sloane_, Nov 17 2009 *)
%o A144124 (Magma) P<x> := PolynomialRing(IntegerRing()); LP_4<x>:=LegendrePolynomial(4); [ Evaluate(LP_4, 2*n+1): n in [0..25] ]; // _Klaus Brockhaus_, Nov 21 2009
%o A144124 (PARI) a(n)=pollegendre(4,n+n+1) \\ _Charles R Greathouse IV_, Oct 25 2011
%Y A144124 Cf. A140870.
%K A144124 nonn,easy
%O A144124 0,2
%A A144124 _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008
%E A144124 Definition corrected by _N. J. A. Sloane_, Nov 17 2009