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]

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

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).

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 64

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)

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