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 A112541 #29 Aug 09 2025 11:23:58
%S A112541 1,2,8,48,400,4390,60624,1013404,19881728,447085170,11319529600,
%T A112541 318298578664,9834869311488,331059072378814,12055438037135360,
%U A112541 472096504892128500,19781301201305534464,882991510898240350666,41828674437875442696192,2095750482492627217639360
%N A112541 a(n) = Sum_{k=0..n} (n-k)! * n^k.
%C A112541 This sequence appears in the calculation of the expectation of the number of runs of an n-faced die, stopping when a face appears for the second time.
%H A112541 G. C. Greubel, <a href="/A112541/b112541.txt">Table of n, a(n) for n = 0..350</a>
%F A112541 a(n) = Sum_{k=0..n} k! * n^(n-k). - _G. C. Greubel_, Jan 12 2022
%p A112541 A112541 := proc(n)
%p A112541 add((n-k)!*n^k,k=0..n) ;
%p A112541 end proc:
%p A112541 seq(A112541(n),n=0..13) ; # _R. J. Mathar_, Dec 16 2015
%t A112541 a[n_]:= Sum[(n-k)!n^k, {k, 0, n}]; Array[a, 17] (* _Robert G. Wilson v_, Dec 22 2005 *)
%o A112541 (Magma) [(&+[Factorial(k)*n^(n-k): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Jan 12 2022
%o A112541 (Sage) [sum(factorial(k)*n^(n-k) for k in (0..n)) for n in (0..20)] # _G. C. Greubel_, Jan 12 2022
%o A112541 (PARI) a(n) = my(A = 1, B = 1); for(k=1, n, B *= n; A = (n-k+1)*A + B); A \\ _Mikhail Kurkov_, Aug 09 2025
%Y A112541 Cf. A000142.
%K A112541 nonn
%O A112541 0,2
%A A112541 _Roger Cuculière_, Dec 17 2005
%E A112541 Corrected and extended by _Robert G. Wilson v_, Dec 22 2005