0,3

a(n) counts permutations of [1,...,n+1] having no substring [k,k+1]. - Len Smiley (smiley(AT)math.uaa.alaska.edu), Oct 13 2001

Also a(n-2) = !n/(n - 1) where !n is the subfactorial of n, A000166(n) - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 18 2002

Also, for n>0, determinant of the tridiagonal n X n matrix M such that M(i,i)=i and for i=1,..,n-1, M(i,i+1)=-1, M(i+1,i)=i. - Mario Catalani (mario.catalani(AT)unito.it), Feb 04 2003

Also, for n>0, maximal permanent of a nonsingular n X n (0,1)-matrix, which is achieved by the matrix with just n-1 0's, all on main diagonal. - Edwin Clark (eclark(AT)math.usf.edu), Oct 28, 2003. For proof see next line.

Proof from Richard Brualdi and Edwin Clark, Nov 15 2003: Let n >=4. Take an n X n (0,1)-matrix A which is nonsingular. It has t >= n-1, 0's, otherwise there will be two rows of all 1's. Let B be the matrix obtained from A by replacing t-(n-1) of A's 0's with 1's. Let D be the matrix with all 1's except for 0's in the first n-1 positions on the diagonal. This matrix is easily seen to be non-singular. Now we have per(A) < = per(B) < = per (D), where the first inequality follows since replacing 0's by 1's cannot decrease the permanent and the second from Corollary 4.4 in the Brualdi et al. reference, which shows that per(D) is the maximum permanent of ANY n X n matrix with n -1 0's. Corollary 4.4 requires n >=4. a(n) for n < 4 can be computed directly.

With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=1 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - Jaap Spies (j.spies(AT)hccnet.nl), Dec 12 2003

Number of fixed-point-free permutations of n+2 that begin with a 2, e.g. for 1234, we have 2143, 2341, 2413, so a(2)=3. Also number of permutations of 2,...,n+2 that have no agreements with 1,...,n+1. E.g. for 123 against permutations of 234, we have 234, 342 and 432. Compare A047920. - Jon Perry (perry(AT)globalnet.co.uk), Jan 23 2004. [This can be proved by the standard argument establishing that d(n+2) = (n+1)(d(n+1)+d(n)) for derangements A000166 (n+1 choices of where 1 goes, then either 1 is in a transposition, or in a cycle of length at least 3, etc.). - D. G. Rogers (drogers(AT)turing.une.edu.au), Aug 28 2006]

Stirling transform of A006252(n+1)=[1,1,2,4,14,38,...] is a(n)=[1,3,11,53,309,...]. - Michael Somos Mar 04 2004

A000255(n+1) is the sequence of numerators of the self-convergents to 1/(e-2); see A096654. - Clark Kimberling (ck6(AT)evansville.edu), Jul 01 2004

Euler's interpretation was "fixedpoint-free permutations beginning with 2" and he listed the terms up to 148329 (although he was blind at the time). - D. E. Knuth, Jan 25 2007

Equals Lim_{k->inf.} A153869^k [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 03 2009]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2009: (Start)

Connection to A002469 (game of mousetrap with n cards): A002469(n) =

(n-2)*A000255(n-1) + A000166(n). (Cf. triangle A159610). (End)

Hankel transform is A059332. [From Paul Barry (pbarry(AT)wit.ie), Apr 22 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

This sequence appears in the analysis of Euler's divergent series 1 - 1! + 2! - 3! + 4! ... by Lacroix, see Hardy. For information about this and related divergent series see A163940.

(End)

Brualdi, Richard A.; Goldwasser, John L.; and Michael, T. S., Maximum permanents of matrices of zeros and ones. J. Combin. Theory Ser. A 47 (1988), 207-245.

R. A. Brualdi and H. J. Ryser: Combinatorial Matrix Theory, Camb. Univ. Press, 1991, Section 7.2, p. 202.

CRC Handbook of Combinatorial Designs, 1996, p. 104.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

G. Kreweras, The number of more or less "regular" permutations, Fib. Quart., 18 (1980), 226-229.

A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 188.

D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16.

M. Rumney and E. J. F. Primrose, A sequence connected with the subfactorial sequence, Note 3207, Math. Gaz. 52 (1968), 381-382.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

L. Euler, "Recherces sur une nouvelle espece des quarres magiques," Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen, 9 (1782), 85-239, on page 235 in section 152. This is paper E530 in Enestrom's index of Euler's works. The sequence appears on page 389 of Euler's Opera Omnia, series I, volume 7. [From D. E. Knuth]

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 179, Table 5.4.

T. D. Noe, Table of n, a(n) for n=0..100

F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 373

G.H. Hardy, Divergent Series , Oxford University Press, 1949. p. 29. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009]

E.g.f.: exp(-x)/(1-x)^2.

a(n)=sum((-1)^k * (n-k+1) * n!/k!, k=0..n) - Len Smiley (smiley(AT)math.uaa.alaska.edu)

Inverse binomial transform of n!. - Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 09 2001

a(n) = floor((1/e)*n!*(n+2)+1/2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 15 2004

Apparently lim n->inf ln(n) - ln(a(n))/n = 1 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 12 2004

a(n)=(n*(n+1)*an(n-1) +(-1)^n)/(n+1),n>=1, a(0)=1. See the Charalambides reference.

a(n) = GAMMA(n+3,-1)*exp(-1)/(n+1) (incomplete Gamma function) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 11 2009]

a(n) = A000166(n) + A000166(n+1).

Cf. A000166.

a(3)=11: 1 3 2 4; 1 4 3 2; 2 1 4 3; 2 4 1 3; 3 2 1 4; 3 2 4 1; 4 1 3 2; 4 2 1 3; 4 3 2 1; 2 4 3 1; 3 1 4 2. The last two correspond to (n-1)*a(n-2) since they contain a [j,n+1,j+1].

c = CoefficientList[Series[Exp[ -z]/(1 - z)^2, {z, 0, 30}], z] For[n = 0, n < 31, n++; Print[c[[n]]*(n - 1)! ]]

Table[Subfactorial[n] + Subfactorial[n + 1], {n, 0, 20}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2009]

(PARI) a(n)=if(n<0, 0, contfracpnqn(matrix(2, n, i, j, j-(i==1)))[1, 1])

(PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(-x+x*O(x^n))/(1-x)^2, n))

(Other) sage: from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2 sage: e = ExtremesOfPermanentsSequence2() sage: it = e.gen(1, 1, 1) sage: [it.next() for i in range(20)] #5 rows.# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]

Row sums of triangle in A046740. A diagonal of triangle in A068106.

Cf. A000153, A000261, A001909, A001910, A090010, A055790, A090012-A090016.

A052655 gives occurrence count for non-singular (0, 1)-matrices with maximal permanent, A089475 number of different values of permanent, A089480 occurrence counts for permanents all non-singular (0, 1)-matrices, A087982, A087983.

A diagonal in triangle A010027.

A153869 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 03 2009]

A159610, A002469 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2009]

Sequence in context: A039302 A074512 A005502 this_sequence A121580 A081367 A156171

Adjacent sequences: A000252 A000253 A000254 this_sequence A000256 A000257 A000258

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).