0,2

a(n) is the number of matchings in the bipartite graph K(n,n). - Sharon Sela (sharonsela(AT)hotmail.com), May 19 2002

Number of 12-avoiding signed permutations in B_n (see Simion ref).

EXP transform of A001048(n) = n! + (n-1)!. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Dec 28 2006

a(n) is also the order of the symmetric inverse semigroup (monoid), I sub n. [From A. Umar (aumarh(AT)squ.edu.om), Sep 09 2008]

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

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

Borwein, D., Rankin, S. and Renner, L. Enumeration of injective partial transformations. Discrete Math. (1989), 73: 291-296. [From A. Umar (aumarh(AT)squ.edu.om), Sep 09 2008]

D. Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.

J. M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995). [From A. Umar (aumarh(AT)squ.edu.om), Sep 09 2008]

Munn, W. D. The characters of the symmetric inverse semigroup. Proc. Cambridge Philos. Soc. 53 (1957), 13-18. [From A. Umar (aumarh(AT)squ.edu.om), Sep 09 2008]

J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.

R. Simion, Combinatorial statistics on type-B analogues of non-crossing partitions and restricted permutations, Electronic J. of Comb. 7 (2000), Art #R9

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

K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009)

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 64

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

Index entries for sequences related to Laguerre polynomials

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 598

a(n) = Sum k!C(n, k)^2, k=0..n. E.g.f.: (1/(1-x))*exp(x/(1-x)). Recurrence: a(n) = 2n*a(n-1) - (n-1)^2*a(n-2).

Sum( (k+n)!^2 / (k+n)!*(k!^2)*exp(1)), k = 0 .. infinity. - Robert G. Wilson v (rgwv(AT)rgwv.com), May 02 2002

a(n) = Sum{m>=0} (-1)^m*A021009(n, m). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 10 2004

a(n)=sum{k=0..n, C(n, k)n!/k!} - Paul Barry (pbarry(AT)wit.ie), May 07 2004

a(n) = Sum[P(n, k)C(n, k) {k=0...n}] a(n) = Sum[n!^2 / k!(n-k)!^2 {k=0...n}] - Ross La Haye (rlahaye(AT)new.rr.com), Sep 20 2004

a(n) = Sum_{k=0..n}(-1)^(n-k)*Stirling1(n, k)*Bell(k+1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 18 2005

Define b(n) by b(0) = 1, b(n) = b(n-1) + 1/n * Sum_{0<=k<n} b(k). Then b(n) = a(n)/n!. - Franklin T. Adams-Watters, Sep 05 2005

Asymptotically, a(n)/n! ~ (1/2)*Pi^(-1/2)*exp(-1/2+2*n^(1/2))/n^(1/4) and so a(n) ~ C*BesselI(0, 2*sqrt(n))*n! with C = exp(-1/2) = .6065306597126334236... - Alec Mihailovs, Sep 06 2005, establishing a conjecture of Franklin T. Adams-Watters.

a(n) = sum {k=0..inf}[binomial(n+k,n)/k! ] * n! / exp(1) - Gottfried Helms (helms(AT)uni-kassel.de), Nov 25 2006

Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation: a(n)=int(x^n*BesselI(0,2*sqrt(x))*exp(-x)/exp(1), x=0..infinity), n=0,1... . From Karol A. Penson (penson(AT)lptl.jussieu.fr) and G. H. E. Duchamp (gduchamp2(AT)free.fr) Jan 09 2007

Table[ n! LaguerreL[ n, -1 ], {n, 0, 12} ].

(PARI) a(n) = suminf ( k=0, binomial(n+k, n)/k! ) / ( exp(1)/n! ) - Gottfried Helms (helms(AT)uni-kassel.de), Nov 25 2006

Cf. A000110, A020556, A069223.

Main diagonal of A088699.

Cf. A000712, A001048.

Sequence in context: A011800 A112916 A145845 this_sequence A111539 A074059 A135882

Adjacent sequences: A002717 A002718 A002719 this_sequence A002721 A002722 A002723

nonn,easy,nice

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

E.g.f. from D. E. Knuth 7/95. 2nd description from R. H. Hardin (rhhardin(AT)att.net) 11/97. 3rd description from wouter.meeussen(AT)pandora.be 6/98.

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 29 2000