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!].
Niu,A short proof of Mathar’s 2014 recurrence conjecture for OEIS A002627, manuscript in preparation
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The Meixner sequence A214615 satisfies the recurrence a(n) - a(n-1) + (n-1)^2 a(n-2) = 0 for n >= 2, proved via its EGF satisfying a linear ODE.
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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!].
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A short proof of Mathar's 2013 recurrence conjecture for the Meixner sequence A214615
The Meixner sequence A214615 satisfies the recurrence a(n) - a(n-1) + (n-1)^2 a(n-2) = 0 for n >= 2, proved via its EGF satisfying a linear ODE.