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

Author

Henry Bottomley, Jun 20 2000

Keywords

• easy
• nonn

Comments

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

Examples

			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
		

References

• 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.

Links

• 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)

Crossrefs

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).

Programs

Haskell

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

Magma

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

Mathematica

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 *)

Formula

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

Extensions

More terms from James Sellers, Jul 04 2000