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A000255 a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
(Formerly M2905 N1166)
78
1, 1, 3, 11, 53, 309, 2119, 16687, 148329, 1468457, 16019531, 190899411, 2467007773, 34361893981, 513137616783, 8178130767479, 138547156531409, 2486151753313617, 47106033220679059, 939765362752547227 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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, 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, 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, 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, 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, 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). - Don Knuth, Jan 25 2007

Equals Lim_{k->inf.} A153869^k. - Gary W. Adamson, Jan 03 2009

Connection to A002469 (game of mousetrap with n cards): A002469(n) = (n-2)*A000255(n-1) + A000166(n). (Cf. triangle A159610). - Gary W. Adamson, Apr 17 2009

Hankel transform is A059332. - Paul Barry, Apr 22 2009

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. - Johannes W. Meijer, Oct 16 2009

a(n), n>=1, enumerates also the ways to distribute n beads, labeled differently from 1 to n, over a set of (unordered) necklaces, excluding necklaces with exactly one bead, and one open cord allowed to have any number of beads. Each beadless necklace as well as the beadless cord contributes a factor 1 in the counting, e.g., a(0):=1*1=1. There are k! possibilities for the cord with k>=0 beads, which means that the two ends of the cord should be considered as fixed, in short: a fixed cord. This produces for a(n) the exponential (aka binomial) convolution of the sequences {n!=A000142(n)} and the subfactorials {A000166(n)}.

See the formula below. Alternatively, the e.g.f. for this problem is seen to be (exp(-x)/(1-x))*(1/(1-x)), namely the product of the e.g.f.s for the subfactorials (from the unordered necklace problem, without necklaces with exactly one bead) and the factorials (from the fixed cord problem). Therefore the recurrence with inputs holds also. a(0):=1. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 02 2010

a(n) = (n-1)a(n-1) + (n-2)a(n-2) gives the same sequence offset by a 1. - Jon Perry, Sep 20 2012

Also, number of reduced 2 X (n+2) Latin rectangles. - A. H. M. Smeets, Nov 03 2013

REFERENCES

M. H. Albert, M. D. Atkinson, and Robert Brignall, Permutation Classes of Polynomial Growth, Annals of Combinatorics 11 (2007) 249-264.

Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From N. J. A. Sloane, Feb 06 2013

B. Balof, H. Jenne, Tilings, Continued Fractions, Derangements, Scramblings, and e, - Journal of Integer Sequences, 17 (2014), #14.2.7.

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.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 179, Table 5.4 and p. 177 (5.1).

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, pp. 263-264. See Table 7.5.1, row 0; also Table 7.6.1, row 0.

L. Euler, "Recherches 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]

Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson Scheme, and Generalized Derangements, International Journal of Combinatorics, Volume 2011, Article ID 539030, 29 pages; doi:10.1155/2011/539030.

H. K. Jenne, Proofs you can count on, Honors Thesis, Math. Dept., Whitman College, 2013; http://www.whitman.edu/mathematics/SeniorProjectArchive/2013/Jenne.pdf

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).

Isaac Sofair, Derangement revisited, Math. Gazette, 97 (No. 540, 2013), 435-440.

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

LINKS

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

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

G. H. Hardy, Divergent Series, Oxford University Press, 1949. p. 29.

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

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

FORMULA

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+1)!. - Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 09 2001

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

a(n) = {(n+2)n!/e}, where {x} denotes the nearest integer. Proposed by Simon Plouffe, March 1993.

Apparently lim n->inf log(n) - log(a(n))/n = 1. - Gerald McGarvey, Jun 12 2004

a(n)=(n*(n+2)*a(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). - Mark van Hoeij, Nov 11 2009

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

Cf. A000166.

If we take b(n) = (-1)^(n+1)*a(n) for n > 0, then for n > 1 the arithmetic mean of the first n terms is -b(n-1). - Franklin T. Adams-Watters, May 20 2010

a(n) = hypergeometric([2,-n],[],1)*(-1)^n = KummerU(2,3+n,-1)*(-1)^n. See the Abramowitz-Stegun handbook (for the reference see e.g. A103921) p. 504, 13.1.10, and for the recurrence p. 507, 13.4.16. - Wolfdieter Lang, May 20 2010

Contribution from Wolfdieter Lang, Jun 02 2010: (Start)

a(n) = n!*(1+sum(sf(n-k)/(n-k)!, k = 0 .. n-2)) with the subfactorials sf(n):= A000166(n) (this follows from the exponential convolution).

a(n) = sf(n+1) + sf(n), n>=0, with sf(n):=A000166(n). (Observation in an e-mail from Gary Detlefs.) (End)

a(n)= 1/(n+1)*floor(((n+1)!+1)/e). - Gary Detlefs, Jul 11 2010

a(n) = (Subfactorial[n+2])/(n+1). - Alexander R. Povolotsky, Jan 26 2011

G.f.: 1/(1-x-2x^2/(1-3x-6x^2/(1-5x-12x^2/(1-7x-20x^2/(1-.../(1-(2n+1)x-(n+1)(n+2)x^2/(1-.. (continued fraction). - Paul Barry, Apr 11 2011

G.f. hypergeom([1,2],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011

E.g.f. 1/E(0) where E(k)=  1 - 2*x/(1 + x/(2 - x - 2/(1 + x*(k+1)/E(k+1)))); (continued fraction 3rd kind, 4-step ). - Sergei N. Gladkovskii, Sep 24 2012

G.f.: S(x)/x - 1/x = Q(0)/x - 1/x where S(x) = sum(k>=0, k!*(x/(1+x))^k  ), Q(k) = 1 + (2*k + 1)*x/( 1 + x - 2*x*(1+x)*(k+1)/(2*x*(k+1) + (1+x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 09 2013

G.f.: 1/Q(0), where Q(k)= 1 + x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013

G.f.: 1/x/Q(0), where Q(k)= 1/x - (2*k+1) - (k+2)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 25 2013

G.f.: (1+x)/(x*Q(0)) - 1/x, where Q(k)= 1 - 2*k*x - x^2*(k + 1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 08 2013

G.f.: 2/x/G(0) - 1/x, where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+1) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013

G.f.: (sum(k>=0, k!*(x/(1+x))^k ) - 1)/x = Q(0)/(2*x) - 1/x, where Q(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1+x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013

G.f.: W(0), where W(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+2)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013

From Peter Bala, Sep 20 2013: (Start)

The sequence b(n) := n!*(n + 2) satisfies the defining recurrence for a(n) but with the starting values b(0) = 2 and b(1) = 3. This leads to the finite continued fraction expansion a(n) = n!*(n+2)*( 1/(2 + 1/(1 + 1/(2 + 2/(3 + ... + (n-1)/n)))) ), valid for n >= 2.

Also a(n) = n!*(n+2)*( sum {k = 0..n} (-1)^k/(k+2)! ). Letting n -> infinity gives the infinite continued fraction expansion 1/e = 1/(2 + 1/(1 + 1/(2 + 2/(3 + ... + (n-1)/(n + ...)))) ) due to Euler. (End)

a(n) = round((n+2)!/e)/(n+1). - Thomas Ordowski, Nov 07 2013

G.f.: G(0)/(1-x), where G(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - (1-x*(1+2*k))*(1-x*(3+2*k))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 05 2014

0 = a(n)*(+a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(+2*a(n+2) - a(n+3)) + a(n+2)*(+a(n+2)) if n>=0. - Michael Somos, May 06 2014

a(n) = hypergeometric([2,-n],[],1)*(-1)^n for n>=1. - Peter Luschny, Sep 20 2014

EXAMPLE

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].

Cord-necklaces problem. For n=4 one considers the following weak two part compositions of 4: (4,0), (2,2), (1,3), and (0,4), where (3,1) does not appear because there are no necklaces with 1 bead. These compositions contribute respectively 4!*1, (binomial(4,2)*2)*sf(2), (binomial(4,1)*1)*sf(3), and 1*sf(4) with the subfactorials sf(n):=A000166(n) (see the necklace comment there). This adds up as 24 + 6*2 + 4*2 + 9 = 53 = a(4). - Wolfdieter Lang, Jun 02 2010

G.f. = 1 + x + 3*x^2 + 11*x^3 + 53*x^4 + 309*x^5 + 2119*x^6 + 16687*x^7 + ...

MAPLE

a := n -> `if`(n=0, 1, hypergeom([2, -n], [], 1))*(-1)^n; seq(round(evalf(a(n), 100)), n=0..19); # Peter Luschny, Sep 20 2014

MATHEMATICA

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}] (* Zerinvary Lajos, Jul 09 2009 *)

RecurrenceTable[{a[n]==n a[n-1]+(n-1)a[n-2], a[0]==1, a[1]==1}, a[n], {n, 20}] (* Harvey P. Dale, May 10 2011 *)

a[ n_] := If[ n < 0, 0, Round[ n! (n + 2) / E]] (* Michael Somos, Jun 01 2013 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ -x] / (1 - x)^2, {x, 0, n}] (* Michael Somos, Jun 01 2013 *)

a[ n_] := If[ n < 0, 0, (-1)^n HypergeometricPFQ[ {- n, 2}, {}, 1]] (* Michael Somos, Jun 01 2013 *)

Rest@ LinearRecurrence[{#1, #1 - 1}, {0, 1}, 21] (* Robert G. Wilson v, Jun 15 2013 *)

PROG

(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))};

(Sage) from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2

e = ExtremesOfPermanentsSequence2()

it = e.gen(1, 1, 1)

[it.next() for i in range(20)]

## [Zerinvary Lajos, May 15 2009]

(Haskell)

a000255 n = a000255_list !! n

a000255_list = 1 : 1 : zipWith (+) zs (tail zs) where

   zs = zipWith (*) [1..] a000255_list

-- Reinhard Zumkeller, Dec 05 2011

CROSSREFS

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.

Cf. A153869 A159610, A002469.

a(n) = A086764(n+1,1).

Sequence in context: A039302 A074512 A005502 * A121580 A224345 A081367

Adjacent sequences:  A000252 A000253 A000254 * A000256 A000257 A000258

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 31 15:25 EDT 2014. Contains 248868 sequences.