A329427 -id:A329427 - cp's OEIS frontend

A056542 a(n) = n*a(n-1) + 1, a(1) = 0.

0, 1, 4, 17, 86, 517, 3620, 28961, 260650, 2606501, 28671512, 344058145, 4472755886, 62618582405, 939278736076, 15028459777217, 255483816212690, 4598708691828421, 87375465144740000, 1747509302894800001, 36697695360790800022, 807349297937397600485

Offset: 1

Henry Bottomley, Jun 20 2000

For n >= 2 also operation count to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of loop repetitions of the j search loop in step L2. - Hugo Pfoertner, Feb 06 2003
More directly: sum over all permutations of length n-1 of the product of the length of the first increasing run by the value of the first position. The recurrence follows from this definition. - Olivier Gérard, Jul 07 2011
This sequence shares divisibility properties with A000522; each of the primes in A072456 divide only a finite number of terms of this sequence. - T. D. Noe, Jul 07 2005
This sequence also represents the number of subdeterminant evaluations when calculation a determinant by Laplace recursive method. - Reinhard Muehlfeld, Sep 14 2010
Also, a(n) equals the number of non-isomorphic directed graphs of n+1 vertices with 1 component, where each vertex has exactly one outgoing edge, excluding loops and cycle graphs. - Stephen Dunn, Nov 30 2019

			a(4) = 4*a(3) + 1 = 4*4 + 1 = 17.
Permutations of order 3 .. Length of first run * First position
123..3*1
132..2*1
213..1*2
231..2*2
312..1*3
321..1*3
a(4) = 3+2+2+4+3+3 = 17. - _Olivier Gérard_, Jul 07 2011

D. E. Knuth: The Art of Computer Programming, Volume 4, Combinatorial Algorithms, Volume 4A, Enumeration and Backtracking. Pre-fascicle 2B, A draft of section 7.2.1.2: Generating all permutations. Available online; see link.

T. D. Noe, Table of n, a(n) for n = 1..100
D. E. Knuth, TAOCP Vol. 4, Pre-fascicle 2b (generating all permutations).
Tom Muller, Prime and Composite Terms in Sloane's Sequence A056542, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.3. [Includes factorizations of a(1) through a(50)]
Hugo Pfoertner, FORTRAN implementation of Knuth's Algorithm L for lexicographic permutation generation.
R. Sedgewick, Permutation generation methods, Computing Surveys, 9 (1977), 137-164.
Sam Wagstaff, Factorizations of a(51) through a(90)

Cf. A079751 (same recursion formula, but starting from a(3)=0), A038155, A038156, A080047, A080048, A080049.
Cf. A007808, A002627.
Equals the row sums of A162995 triangle (n>=2). - Johannes W. Meijer, Jul 21 2009
Cf. A073333, A002627, A185108.
Cf. A329426, A329427.
Cf. A070213 (indices of primes).

a056542 n = a056542_list !! (n-1)
a056542_list = 0 : map (+ 1) (zipWith (*) [2..] a056542_list)
-- Reinhard Zumkeller, Mar 24 2013

[n le 2 select n-1 else n*Self(n-1)+1: n in [1..20]]; // Bruno Berselli, Dec 13 2013

tmp=0; Join[{tmp}, Table[tmp=n*tmp+1, {n, 2, 100}]] (* T. D. Noe, Jul 12 2005 *)
FoldList[ #1*#2 + 1 &, 0, Range[2, 21]] (* Robert G. Wilson v, Oct 11 2005 *)

a(n) = floor((e-2)*n!).
a(n) = A002627(n) - n!.
a(n) = A000522(n) - 2*n!.
a(n) = n! - A056543(n).
a(n) = (n-1)*(a(n-1) + a(n-2)) + 2, n > 2. - Gary Detlefs, Jun 22 2010
1/(e - 2) = 2! - 2!/(1*4) - 3!/(4*17) - 4!/(17*86) - 5!/(86*517) - ... (see A002627 and A185108). - Peter Bala, Oct 09 2013
E.g.f.: (exp(x) - 1 - x) / (1 - x). - Ilya Gutkovskiy, Jun 26 2022

More terms from James Sellers, Jul 04 2000

A329426 Number of non-isomorphic directed graphs where every vertex has outdegree 1, and no self-loops.

Original entry on oeis.org

1, 2, 6, 20, 97, 550, 3794, 29826, 266527, 2649156, 29040865, 347548542, 4509961264, 63050417976, 944767674590, 15103712944100, 256594870255076, 4616238126871328, 87670085904641440, 1752759735606185804, 36796608121601906104, 809312755145598475440, 18609995953274373396982

Offset: 2

Stephen Dunn, Nov 30 2019

nonn

			For n = 2, a(2) = 1 + A329427(2) + A056542(1) = 1 + 0 + 0 = 1, which is the graph A <--> B.
For n = 3, a(3) = 1 + A329427(3) + A056542(2) = 1 + 0 + 1 = 2, which are graphs A --> B <--> C and A --> B --> C --> A.
The middle term is nonzero when there are graphs with more than 1 component.

Stephen Dunn, Table of n, a(n) for n = 2..100

Cf. A329427, A056542.

fun A329427(n: Long): Long = (2L..(n/2)).map { a(it) * a(n-it) }.sum()
fun A056542(n: Long): Long = if (n == 1L) 0 else n * A056542(n-1) + 1
fun a(n: Long): Long = 1 + A329427(n) + A056542(n-1)

a(n) = 1 + A329427(n) + A056542(n-1).

a(n) = 1 + A056542(n-1) + Sum_{2..floor(n/2)} a(i)*a(n-i).

cp's OEIS Frontend

A056542 a(n) = n*a(n-1) + 1, a(1) = 0.

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