A287272 a(n) is the number of zeros of the Laguerre L(n, x) polynomial in the open interval (-1, +1).

0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Author

Michel Lagneau, May 22 2017

Keywords

• nonn

Comments

The Laguerre polynomials are given by the sum: L(n,x) = Sum_{k=0..n} ((-1)^k)/k!*binomial(n,k)*x^k.

The first few Laguerre polynomials are:

L(0,x) = 1,

L(1,x) = -x + 1,

L(2,x) = ( x^2 - 4*x + 2)/2,

L(3,x) = (-x^3 + 9*x^2 - 18*x + 6)/6,

L(4,x) = ( x^4 - 16*x^3 + 72*x^2 - 96*x + 24)/24,

L(5,x) = (-x^5 + 25*x^4 - 200*x^3 + 600*x^2 - 600*x + 120)/120.

The number of occurrences a(n) = 0, 1, 2,.. is given by the sequence {2, 6, 11, 16, 21, ...}.

Examples

			a(3) = 1 because the zeros of L(3,x) = (-x^3 + 9*x^2 - 18*x + 6)/6 are x1=.4157745568..., x2=2.294280360... and x3=6.289945083... with the root x1 in the open interval (-1, +1). Hence, a(3) = 1.
		

Links

• Eric Weisstein's World of Mathematics, The World of Mathematics: Laguerre Polynomial

Crossrefs

Cf. A066667.

Programs

Maple

for n from 0 to 90 do:it:=0:
y:=[fsolve(expand(LaguerreL(n,x)),x,real)]:for m from 1 to nops(y) do:if abs(y[m])<1 then it:=it+1:else fi:od: printf(`%d, `, it):od: