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

%I A287272 #9 Feb 16 2025 08:33:46
%S A287272 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,
%T A287272 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,
%U A287272 5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A287272 a(n) is the number of zeros of the Laguerre L(n, x) polynomial in the open interval (-1, +1).
%C A287272 The Laguerre polynomials are given by the sum: L(n,x) = Sum_{k=0..n} ((-1)^k)/k!*binomial(n,k)*x^k.
%C A287272 The first few Laguerre polynomials are:
%C A287272 L(0,x) =   1,
%C A287272 L(1,x) =  -x + 1,
%C A287272 L(2,x) = ( x^2 - 4*x + 2)/2,
%C A287272 L(3,x) = (-x^3 + 9*x^2 - 18*x + 6)/6,
%C A287272 L(4,x) = ( x^4 - 16*x^3 + 72*x^2 - 96*x + 24)/24,
%C A287272 L(5,x) = (-x^5 + 25*x^4 - 200*x^3 + 600*x^2 - 600*x + 120)/120.
%C A287272 The number of occurrences a(n) = 0, 1, 2,.. is given by the sequence {2, 6, 11, 16, 21, ...}.
%H A287272 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">The World of Mathematics: Laguerre Polynomial</a>
%e A287272 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.
%p A287272 for n from 0 to 90 do:it:=0:
%p A287272 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:
%Y A287272 Cf. A066667.
%K A287272 nonn
%O A287272 0,9
%A A287272 _Michel Lagneau_, May 22 2017