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A173227 Partial sums of A000262.

%I A173227 #14 Mar 09 2021 20:14:40
%S A173227 1,2,5,18,91,592,4643,42276,436629,5033182,63974273,888047414,
%T A173227 13358209647,216334610860,3751352135263,69325155322184,
%U A173227 1359759373992105,28206375825238458,616839844140642301,14181213537729200474,341879141423814854915,8623032181189674581256
%N A173227 Partial sums of A000262.
%C A173227 Partial sums of the number of "sets of lists": number of partitions of {1,..,n} into any number of lists, where a list means an ordered subset. The subsequence of primes begins: 2, 5, 4643, 616839844140642301.
%H A173227 Alois P. Heinz, <a href="/A173227/b173227.txt">Table of n, a(n) for n = 0..444</a>
%H A173227 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A173227 From _Vaclav Kotesovec_, Oct 25 2016: (Start)
%F A173227 a(n) = 2*n*a(n-1) - (n^2 - n + 1)*a(n-2) + (n-2)*(n-1)*a(n-3).
%F A173227 a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/sqrt(2) * (1 - 5/(48*sqrt(n))).
%F A173227 (End)
%F A173227 a(n) = 1 + Sum_{j=1..n} j!*( LaguerreL(j,-1) - LaguerreL(j-1,-1) ). - _G. C. Greubel_, Mar 09 2021
%e A173227 a(20) = 1 + 1 + 3 + 13 + 73 + 501 + 4051 + 37633 + 394353 + 4596553 + 58941091 + 824073141 + 12470162233 + 202976401213 + 3535017524403 + 65573803186921 + 1290434218669921 + 26846616451246353 + 588633468315403843 + 13564373693588558173 + 327697927886085654441.
%p A173227 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A173227        b(n-j)*j!*binomial(n-1, j-1), j=1..n))
%p A173227     end:
%p A173227 a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
%p A173227 seq(a(n), n=0..25);  # _Alois P. Heinz_, May 11 2016
%t A173227 With[{m = 25}, CoefficientList[Exp[x/(1-x)] + O[x]^m, x] Range[0, m-1]!// Accumulate] (* _Jean-François Alcover_, Nov 21 2020 *)
%t A173227 Table[1 +Sum[j!*(LaguerreL[j, -1] -LaguerreL[j-1, -1]), {j,n}], {n,0,30}] (* _G. C. Greubel_, Mar 09 2021 *)
%o A173227 (Sage) [1 + sum(factorial(j)*(gen_laguerre(j,0,-1) - gen_laguerre(j-1,0,-1)) for j in (1..n)) for n in (0..30)] # _G. C. Greubel_, Mar 09 2021
%o A173227 (Magma)
%o A173227 l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
%o A173227 [n eq 0 select 1 else 1 + (&+[ Factorial(j)*( l(j,-1) - l(j-1,-1) ): j in [1..n]]): n in [0..25]]; // _G. C. Greubel_, Mar 09 2021
%Y A173227 Cf. A000262, A001263, A001700, A002868, A008297, A052852, A066668, A111596,.
%K A173227 nonn
%O A173227 0,2
%A A173227 _Jonathan Vos Post_, Feb 13 2010