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I recently learned about the generalized Derangement problem. And I know how to use Laguerre polynomials to find the general term. But it seems difficult to write programs to calculate definite integrals.

$$(-1)^{n_1+n_2+...+n_k}\int\limits_0^\infty e^{-x}\prod_{i=1}^k \{ L_{ni}(x)\} \ dx $$

So I wonder if there is a way to find the recursive sequence of the generalized Derangement, so that the Time complexity of programming is much lower.For example,the Derangement problem's recursive sequence is

$$f(n)=(n-1)*(f(n-2)+f(n-1))$$

And in this questionlink,the recursive sequence is

$(n-3)*(3*n^3 - 36*n^2 + 137*n - 162)*a(n) = (n-5)*(3*n^3 - 27*n^2 + 71*n - 50)*a(n-1) + (n-2)*(3*n^5 - 45*n^4 + 248*n^3 - 606*n^2 + 608*n - 156)*a(n-2) - 2*(n-3)*(3*n^3 - 28*n^2 + 87*n - 94)*a(n-3) + 2*(3*n^5 - 51*n^4 + 334*n^3 - 1060*n^2 + 1650*n - 1028)*a(n-4) - 4*(n-4)*(n^2 + n - 14)*a(n-5) - 4*(n-5)*(n-4)*(n-2)*(3*n^3 - 27*n^2 + 74*n - 58)*a(n-6)$

I see this recursive sequence in $OEISA003471$,which is provided by Vaclav Kotesovec.

So is there any normal way to find the recursive sequence of the generalized Derangement problem?I can't find any information on this matter

ame
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  • The code in my answer to https://mathematica.stackexchange.com/questions/285008/ can be used to find the recursion as in A003471. – Somos Jun 15 '23 at 16:57
  • @Somos TY,sir.I get the right answer by running your code.But it seems that the coefficient of k is a bit large. – ame Jun 16 '23 at 14:13
  • @Somos TY!!!By supplying more sequence values I get the concise and clear answer.Sincere respect to you – ame Jun 17 '23 at 06:51

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