A329426 -id:A329426 - 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

A329427 Number of directed graphs of n vertices with more than 1 component and outdegree 1.

Original entry on oeis.org

1, 2, 10, 32, 173, 864, 5876, 42654, 369352, 3490396, 37205377, 431835570, 5488938513, 75253166882, 1111054042385, 17529435042906, 294620759901439, 5250432711385802, 98912760811106081, 1963457208200874954, 40962100714228585825, 895889161265034629994, 20497593840242211891900

Offset: 4

Stephen Dunn, Nov 30 2019

nonn

a(n) gives the number of unique ways a directed graph of n vertices with outdegree 1 can be broken into smaller components of size >= 2. It can be generalized to higher degree by replacing A329426 in the formula with a suitable counting function.

			a(4) = A329426(2)*A329426(2) = 1*1 = 1, which represents the graph
  V <--> V
  V <--> V.
a(5) = A329426(2)*A329426(3) = 1*2 = 2, which represents the two possible graphs of size 3 (V --> V <--> V, etc.) paired with V <--> V.
a(6) = A329426(2)*A329426(4) + A329426(3)*A329426(3) = 1*6 + 2*2 = 10.

Stephen Dunn, Table of n, a(n) for n = 4..100

Cf. A329426, A056542.

fun A056542(n: Long): Long = if (n == 1L) 0 else n * A056542(n-1) + 1
fun A329426(n: Long): Long = 1 + a(n) + A056542(n-1)
fun a(n: Long): Long = (2L..(n/2)).map { A329426(it) * A329426(n-it) }.sum()

a(n) = Sum_{i=2..floor(n/2)} A329426(i) * A329426(n-i).

Term a(26) corrected by Sidney Cadot, Jan 06 2023.

cp's OEIS Frontend

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

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