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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The sequence A001711 satisfies a(n) - (2n+5)a(n-1) + (n+2)^2 a(n-2) = 0 for n >= 2, shown by substituting the closed form (1/4)(n+3)! (2 H_{n+3} - 3) and verifying that harmonic coefficients and constant terms both cancel to zero.
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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 2020 recurrence conjecture for the generalized-Stirling sequence A001711
The sequence A001711 satisfies a(n) - (2n+5)a(n-1) + (n+2)^2 a(n-2) = 0 for n >= 2, shown by substituting the closed form (1/4)(n+3)! (2 H_{n+3} - 3) and verifying that harmonic coefficients and constant terms both cancel to zero.