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

0,2

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..444

Index entries for sequences related to Laguerre polynomials

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A020025 A113715 A227094 * A099556 A057864 A320154

Adjacent sequences: A173224 A173225 A173226 * A173228 A173229 A173230

Keyword

nonn

Author

Jonathan Vos Post, Feb 13 2010

Status

approved