This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A288268 #42 May 27 2023 02:34:42
%S A288268 1,0,1,4,21,136,1045,9276,93289,1047376,12975561,175721140,2581284541,
%T A288268 40864292184,693347907421,12548540320876,241253367679185,
%U A288268 4909234733857696,105394372192969489,2380337795595885156,56410454014314490981,1399496554158060983080
%N A288268 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)*x^k/k).
%H A288268 Seiichi Manyama, <a href="/A288268/b288268.txt">Table of n, a(n) for n = 0..444</a>
%H A288268 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A288268 a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} (k-1)*a(n-k)/(n-k)! for n > 0.
%F A288268 E.g.f.: (1 - x) * exp(x/(1 - x)). - _Ilya Gutkovskiy_, Jul 27 2020
%F A288268 a(n) = (n!/(n-1))*( 2*LaguerreL(n-1, -1) - LaguerreL(n, -1) ) with a(0) = 1, a(1) = 0. - _G. C. Greubel_, Mar 10 2021
%F A288268 a(n) ~ n^(n - 3/4) * exp(-1/2 + 2*sqrt(n) - n) / sqrt(2) * (1 - 65/(48*sqrt(n))). - _Vaclav Kotesovec_, Mar 10 2021, minor term corrected Dec 01 2021
%F A288268 From _Peter Luschny_, Feb 20 2022: (Start)
%F A288268 a(n) = n! * Sum_{k=0..n} (-1)^k * LaguerreL(n-k, k-1, -1).
%F A288268 a(n) = 2*(n - 1)*a(n - 1) - (n^2 - 4*n + 3)*a(n - 2) for n >= 3. (End)
%F A288268 From _Peter Bala_, May 26 2023: (Start)
%F A288268 a(n) = Sum_{k = 0..n} |Stirling1(n,k)|*A000296(k) (follows from the fundamental theorem of Riordan arrays).
%F A288268 Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is purely periodic with the period dividing k. For example, modulo 7 we obtain the purely periodic sequence [1, 0, 1, 4, 0, 3, 2, 1, 0, 1, 4, 0, 3, 2, ...] of period 7. Cf. A047974. (End)
%F A288268 For n>1, a(n) = (2*n*A002720(n-1) - A002720(n))/(n-1). - _Vaclav Kotesovec_, May 27 2023
%p A288268 a := proc(n) option remember; if n < 3 then [1, 0, 1][n+1] else
%p A288268 -(n^2 - 4*n + 3)*a(n - 2) + (2*n - 2)*a(n - 1) fi end:
%p A288268 seq(a(n), n = 0..21); # _Peter Luschny_, Feb 20 2022
%t A288268 Table[If[n<2, 1-n, (n!/(n-1))*(2*LaguerreL[n-1, -1] - LaguerreL[n, -1])], {n, 0, 30}] (* _G. C. Greubel_, Mar 10 2021 *)
%o A288268 (PARI) {a(n) = n!*polcoeff(exp(sum(k=1, n, (k-1)*x^k/k)+x*O(x^n)), n)}
%o A288268 (Magma)
%o A288268 l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;
%o A288268 [1,0]cat[(Factorial(n)/(n-1))*(2*l(n-1,0,-1) - l(n,0,-1)): n in [2..30]]; // _G. C. Greubel_, Mar 10 2021
%o A288268 (Sage) [1-n if n<2 else (factorial(n)/(n-1))*(2*gen_laguerre(n-1,0,-1) - gen_laguerre(n,0,-1)) for n in (0..30)] # _G. C. Greubel_, Mar 10 2021
%Y A288268 Cf. A000296, A002720, A008275, A009940, A052852, A052887, A059114, A288269.
%K A288268 nonn,easy
%O A288268 0,4
%A A288268 _Seiichi Manyama_, Oct 20 2017