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%I A002720 M1795 N0708
%S A002720 1,2,7,34,209,1546,13327,130922,1441729,17572114,234662231,3405357682,
%T A002720 53334454417,896324308634,16083557845279,306827170866106,
%U A002720 6199668952527617,132240988644215842,2968971263911288999
%N A002720 Number of partial permutations of an n-set; number of n X n binary matrices 
               with at most one 1 in each row and column.
%C A002720 a(n) is the number of matchings in the bipartite graph K(n,n). - Sharon 
               Sela (sharonsela(AT)hotmail.com), May 19 2002
%C A002720 Number of 12-avoiding signed permutations in B_n (see Simion ref).
%C A002720 EXP transform of A001048(n) = n! + (n-1)!. - Franklin T. Adams-Watters 
               (FrankTAW(AT)Netscape.net), Dec 28 2006
%C A002720 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]
%D A002720 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002720 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002720 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 A002720 D. Castellanos, A generalization of Binet's formula and some of its consequences, 
               Fib. Quart., 27 (1989), 424-438.
%D A002720 J. M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, 
               (1995). [From A. Umar (aumarh(AT)squ.edu.om), Sep 09 2008]
%D A002720 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]
%D A002720 J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, 
               Paris, 1933, p. 78.
%D A002720 R. Simion, Combinatorial statistics on type-B analogues of non-crossing 
               partitions and restricted permutations, Electronic J. of Comb. 7 
               (2000), Art #R9
%H A002720 T. D. Noe, <a href="b002720.txt">Table of n, a(n) for n=0..100</a>
%H A002720 K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, 
               <a href="http://arxiv.org/abs/0904.0369">Laguerre-type derivatives: 
               Dobinski relations and combinatorial identities</a>, J. Math. Phys. 
               vol. 50, 083512 (2009)
%H A002720 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=64">
               Encyclopedia of Combinatorial Structures 64</a>
%H A002720 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</
               a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A002720 <a href="Sindx_La.html#Laguerre">Index entries for sequences related 
               to Laguerre polynomials</a>
%H A002720 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               598
%F A002720 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).
%F A002720 Sum( (k+n)!^2 / (k+n)!*(k!^2)*exp(1)), k = 0 .. infinity. - Robert G. 
               Wilson v (rgwv(AT)rgwv.com), May 02 2002
%F A002720 a(n) = Sum{m>=0} (-1)^m*A021009(n, m). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 10 2004
%F A002720 a(n)=sum{k=0..n, C(n, k)n!/k!} - Paul Barry (pbarry(AT)wit.ie), May 07 
               2004
%F A002720 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
%F A002720 a(n) = Sum_{k=0..n}(-1)^(n-k)*Stirling1(n, k)*Bell(k+1). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Mar 18 2005
%F A002720 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
%F A002720 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.
%F A002720 a(n) = sum {k=0..inf}[binomial(n+k,n)/k! ] * n! / exp(1) - Gottfried 
               Helms (helms(AT)uni-kassel.de), Nov 25 2006
%F A002720 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
%t A002720 Table[ n! LaguerreL[ n, -1 ], {n, 0, 12} ].
%o A002720 (PARI) a(n) = suminf ( k=0, binomial(n+k,n)/k! ) / ( exp(1)/n! ) - Gottfried 
               Helms (helms(AT)uni-kassel.de), Nov 25 2006
%Y A002720 Cf. A000110, A020556, A069223.
%Y A002720 Main diagonal of A088699.
%Y A002720 Cf. A000712, A001048.
%Y A002720 Sequence in context: A011800 A112916 A145845 this_sequence A111539 A074059 
               A135882
%Y A002720 Adjacent sequences: A002717 A002718 A002719 this_sequence A002721 A002722 
               A002723
%K A002720 nonn,easy,nice
%O A002720 0,2
%A A002720 N. J. A. Sloane (njas(AT)research.att.com).
%E A002720 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.
%E A002720 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 29 2000

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