The exponential generating function F(x) = -exp(-x/(1-2x))/(1-2x) for a(n) = -n! 2^n L_n(1/2) satisfies the ODE (1-2x)^2 F'(x) = (1-4x) F(x), from which the recurrence a(n) + (-4n+3)a(n-1) + 4(n-1)^2 a(n-2) = 0 follows by extracting [x^n/n!].
Fried,Proofs of several conjectures from the OEIS, J
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A short proof of Mathar's 2013 recurrence conjecture for the Laguerre sequence~A025166
The exponential generating function F(x) = -exp(-x/(1-2x))/(1-2x) for a(n) = -n! 2^n L_n(1/2) satisfies the ODE (1-2x)^2 F'(x) = (1-4x) F(x), from which the recurrence a(n) + (-4n+3)a(n-1) + 4(n-1)^2 a(n-2) = 0 follows by extracting [x^n/n!].