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 A059115 #21 Feb 23 2021 18:03:40
%S A059115 1,2,9,58,485,4986,60877,861554,13878153,250854130,5030058161,
%T A059115 110837000682,2662669300909,69270266115818,1940260799150325,
%U A059115 58220372514830626,1863293173842259217,63356877145370671074
%N A059115 Expansion of e.g.f.: ((1-x)/(1-2*x))*exp(x/(1-x)).
%C A059115 L'(n,i) are unsigned Lah numbers (Cf. A008297): L'(n,i) = (n!/i!)*binomial(n-1,i-1) for i >= 1, L'(0,0) = 1, L'(n,0) = 0 for n > 0.
%H A059115 G. C. Greubel, <a href="/A059115/b059115.txt">Table of n, a(n) for n = 0..250</a>
%H A059115 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A059115 Sum_{m=0..n} Sum_{i=0..n} L'(n, i)*Product_{j=1..m} (i-j+1).
%F A059115 Given g.f. A(x), then g.f. A000522 = A(x/(1+x)). - _Michael Somos_, Aug 03 2006
%F A059115 a(n) = n!*Sum_{k=0..n} LaguerreL(n-k, k-1, -1). - _G. C. Greubel_, Feb 23 2021
%F A059115 a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n-1). - _Vaclav Kotesovec_, Feb 23 2021
%e A059115 (1-x)/(1-2*x)*exp(x/(1-x)) = 1 + 2*x + 9/2*x^2 + 29/3*x^3 + 485/24*x^4 + 831/20*x^5 + ...
%p A059115 s := series((1-x)/(1-2*x)*exp(x/(1-x)), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:
%t A059115 With[{nn=20},CoefficientList[Series[(1-x)/(1-2x) Exp[x/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 18 2020 *)
%t A059115 Table[n!*Sum[LaguerreL[n-k, k-1, -1], {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Feb 23 2021 *)
%o A059115 (PARI) {a(n)=if(n<0, 0, n!*polcoeff( (1-x)/(1-2*x)*exp(x/(1-x)+x*O(x^n)), n))} /* _Michael Somos_, Aug 03 2006 */
%o A059115 (PARI) {a(n)=local(A); if(n<0,0, n++; A=vector(n); A[n]=1; for(k=1,n-1, A[n-k]=1; if(k>1, A[n-k+1]=A[n-k+2]); for(i=n-k+1,n, A[i]=A[i-1]+k*A[i])); A[n])} /* _Michael Somos_, Aug 03 2006 */
%o A059115 (PARI) a(n) = n!*sum(k=0, n, pollaguerre(n-k, k-1, -1)); \\ _Michel Marcus_, Feb 23 2021
%o A059115 (Sage) [factorial(n)*sum( gen_laguerre(n-k, k-1, -1) for k in (0..n) ) for n in (0..30)] # _G. C. Greubel_, Feb 23 2021
%o A059115 (Magma) [Factorial(n)*(&+[Evaluate(LaguerrePolynomial(n-k, k-1), -1) : k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 23 2021
%Y A059115 Cf. A001861, A049020, A052897, A059099, A059110.
%Y A059115 Row sums of A059114.
%K A059115 easy,nonn
%O A059115 0,2
%A A059115 _Vladeta Jovovic_, Jan 06 2001
%E A059115 Definition clarified by _Harvey P. Dale_, Jul 18 2020