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

1, 321, 2641, 10321, 28401, 63601, 124321, 220641, 364321, 568801, 849201, 1222321, 1706641, 2322321, 3091201, 4036801, 5184321, 6560641, 8194321, 10115601, 12356401, 14950321, 17932641, 21340321, 25212001, 29588001

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

Keywords

• nonn
• easy

Comments

Legendre polynomial LP_4(x) = (35*x^4 - 30*x^2 + 3)/8. - Klaus Brockhaus, Nov 21 2009

Links

• Eric W. Weisstein, Legendre Polynomial.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A140870.

Programs

Magma

P := PolynomialRing(IntegerRing()); LP_4:=LegendrePolynomial(4); [ Evaluate(LP_4, 2*n+1): n in [0..25] ]; // Klaus Brockhaus, Nov 21 2009

Mathematica

Table[LegendreP[4,2n+1],{n,0,50}] (* N. J. A. Sloane, Nov 17 2009 *)

PARI

a(n)=pollegendre(4,n+n+1) \\ Charles R Greathouse IV, Oct 25 2011

Formula

From Klaus Brockhaus, Nov 21 2009: (Start)

a(n) = 70*n^4 + 140*n^3 + 90*n^2 + 20*n + 1.

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.

G.f.: (1 + 316*x + 1046*x^2 + 316*x^3 + x^4)/(1-x)^5. (End)

Extensions

Definition corrected by N. J. A. Sloane, Nov 17 2009