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A140870 8*P_4(2n), 8 times the Legendre Polynomial of order 4 at 2n.

%I A140870 #23 Feb 16 2025 08:33:08
%S A140870 3,443,8483,44283,141443,347003,721443,1338683,2286083,3664443,
%T A140870 5588003,8184443,11594883,15973883,21489443,28323003,36669443,
%U A140870 46737083,58747683,72936443,89552003,108856443,131125283,156647483,185725443,218675003,255825443
%N A140870 8*P_4(2n), 8 times the Legendre Polynomial of order 4 at 2n.
%H A140870 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial.</a>
%H A140870 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A140870 Legendre polynomial LP_4(x) = (35*x^4-30*x^2+3)/8. - _Klaus Brockhaus_, Nov 21 2009
%F A140870 From _Klaus Brockhaus_, Nov 21 2009: (Start)
%F A140870 a(n) = 560*n^4-120*n^2+3.
%F A140870 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4)+13440 for n > 3; a(0)=3, a(1)=443, a(2)=8483, a(3)=44283.
%F A140870 G.f.: (3+428*x+6298*x^2+6268*x^3+443*x^4)/(1-x)^5. (End)
%p A140870 A140870 := proc(n)
%p A140870         8*orthopoly[P](4,2*n) ;
%p A140870 end proc: # _R. J. Mathar_, Oct 24 2011
%t A140870 Table[8 LegendreP[4,2n],{n,0,50}]
%t A140870 LinearRecurrence[{5, -10, 10, -5, 1}, {3, 443, 8483, 44283, 141443}, 30] (* _Vincenzo Librandi_, Oct 04 2015 *)
%o A140870 (Magma)
%o A140870 P<x> := PolynomialRing(IntegerRing());
%o A140870 LP4:=LegendrePolynomial(4);
%o A140870 [ Evaluate(8*LP4, 2*n): n in [0..26] ]; // _Klaus Brockhaus_, Nov 18 2009
%o A140870 (PARI) {for(n=0, 26, print1(subst(8*pollegendre(4), x, 2*n), ","))} \\ _Klaus Brockhaus_, Nov 21 2009
%o A140870 (Magma) [560*n^4 - 120*n^2 + 3: n in [0..30]]; // _Vincenzo Librandi_, Oct 04 2015
%Y A140870 Cf. A144124.
%K A140870 nonn,easy
%O A140870 0,1
%A A140870 _N. J. A. Sloane_, Nov 17 2009