A173227 Partial sums of A000262.

1, 2, 5, 18, 91, 592, 4643, 42276, 436629, 5033182, 63974273, 888047414, 13358209647, 216334610860, 3751352135263, 69325155322184, 1359759373992105, 28206375825238458, 616839844140642301, 14181213537729200474, 341879141423814854915, 8623032181189674581256

Offset: 0

Author

Jonathan Vos Post, Feb 13 2010

Keywords

• nonn

Comments

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.

Examples

			a(20) = 1 + 1 + 3 + 13 + 73 + 501 + 4051 + 37633 + 394353 + 4596553 + 58941091 + 824073141 + 12470162233 + 202976401213 + 3535017524403 + 65573803186921 + 1290434218669921 + 26846616451246353 + 588633468315403843 + 13564373693588558173 + 327697927886085654441.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..444
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A000262, A001263, A001700, A002868, A008297, A052852, A066668, A111596,.

Programs

Magma

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[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

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
       b(n-j)*j!*binomial(n-1, j-1), j=1..n))
    end:
a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

Mathematica

With[{m = 25}, CoefficientList[Exp[x/(1-x)] + O[x]^m, x] Range[0, m-1]!// Accumulate] (* Jean-François Alcover, Nov 21 2020 *)
Table[1 +Sum[j!*(LaguerreL[j, -1] -LaguerreL[j-1, -1]), {j,n}], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

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

Formula

From Vaclav Kotesovec, Oct 25 2016: (Start)

a(n) = 2*n*a(n-1) - (n^2 - n + 1)*a(n-2) + (n-2)*(n-1)*a(n-3).

a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/sqrt(2) * (1 - 5/(48*sqrt(n))).

(End)

a(n) = 1 + Sum_{j=1..n} j!*( LaguerreL(j,-1) - LaguerreL(j-1,-1) ). - G. C. Greubel, Mar 09 2021