A289192 -id:A289192 - cp's OEIS frontend

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.

1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114, 234662231, 3405357682, 53334454417, 896324308634, 16083557845279, 306827170866106, 6199668952527617, 132240988644215842, 2968971263911288999, 69974827707903049154, 1727194482044146637521, 44552237162692939114282

Offset: 0

N. J. A. Sloane

a(n) is also the total number of increasing subsequences of all permutations of [1..n] (see Lifschitz and Pittel). - N. J. A. Sloane, May 06 2012
a(n) = A000142 + A001563 + A001809 + A001810 + A001811 + A001812 + ... these sequences respectively give the number of increasing subsequences of length i for i=0,1,2,... in all permutations of [1..n]. - Geoffrey Critzer, Jan 17 2013
a(n) is also the number of matchings in the complete bipartite graph K(n,n). - Sharon Sela (sharonsela(AT)hotmail.com), May 19 2002
a(n) is also the number of 12-avoiding signed permutations in B_n (see Simion ref).
a(n) is also the order of the symmetric inverse semigroup (monoid) I_n. - A. Umar, Sep 09 2008
EXP transform of A001048(n) = n! + (n-1)!. - Franklin T. Adams-Watters, Dec 28 2006
From Peter Luschny, Mar 27 2011: (Start)
Let B_{n}(x) = Sum_{j>=0} exp(j!/(j-n)!*x-1)/j!; then a(n) = 2! [x^2] Taylor(B_{n}(x)), where [x^2] denotes the coefficient of x^2 in the Taylor series for B_{n}(x).
a(n) is column 2 of the square array representation of A090210. (End)
a(n) is the Hosoya index of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Jul 09 2011
a(n) is also number of non-attacking placements of k rooks on an n X n board, summed over all k >= 0. - Vaclav Kotesovec, Aug 28 2012
Also the number of vertex covers and independent vertex sets in the n X n rook graph. - Eric W. Weisstein, Jan 04 2013
a(n) is the number of injective functions from subsets of [n] to [n] where [n]={1,2,...,n}. For a subset D of size k, there are n!/(n-k)! injective functions from D to [n]. Summing over all subsets, we obtain a(n) = Sum_{k=0..n} C(n,k)*n!/(n-k)! = Sum_{k=0..n} k!*C(n,k)^2. - Dennis P. Walsh, Nov 16 2015
Also the number of cliques in the n X n rook complement graph. - Eric W. Weisstein, Sep 14 2017
a(n)/n! is the expected value of the n-th term of Ulam's "history-dependent random sequence". See Kac (1989), Eq.(2). - N. J. A. Sloane, Nov 16 2019
a(2*n) is odd and a(2*n+1) is even for all n. More generally, for each positive integer k, a(n+k) == a(n) (mod k) for all n. It follows that for each positive integer k, the sequence obtained by reducing a(n) modulo k is periodic, with period dividing k. Various divisibility properties of the sequence follow from this: for example, a(7*n+2) == 0 (mod 7), a(11*n+4) == 0 (mod 11), a(17*n+3) == 0 (mod 17) and a(19*n+4) == 0 (mod 19). - Peter Bala, Nov 07 2022
Conjecture: a(n)*k is the sum of the largest parts in all integer partitions containing their own first differences with n + 1 parts and least part k. - John Tyler Rascoe, Feb 28 2024

			G.f. = 1 + 2*x + 7*x^2 + 34*x^3 + 209*x^4 + 1546*x^5 + 13327*x^6 + 130922*x^7 + ... - _Michael Somos_, Jul 31 2018

J. M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995). [From A. Umar, Sep 09 2008]
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
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).
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 356.

Alois P. Heinz, Table of n, a(n) for n = 0..443 (first 101 terms from T. D. Noe)
Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Ramified inverse and planar monoids, arXiv:2210.17461 [math.RT], 2022.
A. I. Aptekarev, On linear forms containing the Euler constant, arXiv:0902.1768 [math.NT], 2009.
T. Banica, The algebraic structure of quantum partial isometries, arXiv:1411.0577 [math.OA], 2014-2015.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy, and D. Gouyou-Beauchamps, Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Teo Banica, Algebraic invariants of truncated Fourier matrices, arXiv preprint arXiv:1401.5023 [math.QA], 2014.
D. Borwein, S. Rankin, and L. Renner, Enumeration of injective partial transformations, Discrete Math. (1989), 73: 291-296. [From A. Umar, Sep 09 2008]
D. Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.
Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024. See p. 14.
Fabio Deelan Cunden, Jakub Czartowski, Giovanni Gramegna, and A. de Oliveira Junior, Relative volume of comparable pairs under semigroup majorization, arXiv:2410.23196 [math-ph], 2024. See pp. 4, 15.
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 598.
Olof Giselsson, The Universal C*-Algebra of the Quantum Matrix Ball and its Irreducible *-Representations, arXiv:1801.10608 [math.QA], 2018.
J. Godbout, Combinatorial Properties of the Mirabolic RSK Algorithm, Thesis presented to The Faculty of the Graduate College of The University of Vermont, May 2013.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 64.
Z. Janelidze, F. van Niekerk, and J. Viljoen, What is the maximal connected partial symmetry index of a connected graph of a given size?, arXiv:2502.00704 [math.CO], 2025. See p. 3.
Mark Kac, A history-dependent random sequence defined by Ulam, Advances in Applied Mathematics 10.3 (1989): 270-277.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 219.
Vaclav Kotesovec, Too many errors around coefficient C1 in asymptotic of sequence A002720, Sep 28 2012. [The bug in program Mathematica was fixed in version 10.2.0.0 / July 2015. _Vaclav Kotesovec_, Jul 25 2015]
V. Lifschitz and P. Pittel, The number of increasing subsequences of the random permutation J. Combin. Theory Ser. A 31 (1981), no. 1, 1--20. MR0626437 (84e:05012)
Mathematica Stack Exchange, Wrong Limit with LaguerreL, May 22 2015
W. D. Munn, The characters of the symmetric inverse semigroup, Proc. Cambridge Philos. Soc. 53 (1957), 13-18. [From A. Umar, Sep 09 2008]
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.
John Riordan, Letter, Apr 28 1976.
John Riordan, Letter to N. J. A. Sloane, Oct 01 1978.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles. (Annotated scans of some selected pages)
R. Simion, Combinatorial statistics on type-B analogues of non-crossing partitions and restricted permutations, Electronic J. of Comb. 7 (2000), Art #R9.
A. Umar, Some combinatorial problems in the theory of symmetric ..., Algebra Disc. Math. 9 (2010) 115-126.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
D. P. Walsh, Notes on functions from subsets of {1,2,...,n} into {1,2,...,n}.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Matching.
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Eric Weisstein's World of Mathematics, Rook Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for sequences related to Laguerre polynomials

Main diagonal of A088699. Column of A283500. Row sums of A144084.
Cf. A000110, A020556, A069223, A000712, A001048, A090210, A002793, A073003, A000262.
Column k=1 of A289192.
Cf. A160617, A160618.
Cf. A364673.

[Factorial(n)*Evaluate(LaguerrePolynomial(n), -1): n in [0..25]]; // G. C. Greubel, Aug 11 2022

A002720 := proc(n) exp(-x)*n!*hypergeom([n+1], [1], x); simplify(subs(x=1, %)) end: seq(A002720(n), n=0..25); # Peter Luschny, Mar 30 2011
A002720 := proc(n)
    option remember;
    if n <= 1 then
        n+1 ;
    else
        2*n*procname(n-1)-(n-1)^2*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Mar 09 2017

Table[n! LaguerreL[n, -1], {n, 0, 25}]
Table[(-1)^n*HypergeometricU[-n, 1, -1], {n, 0, 25}] (* Jean-François Alcover, Jul 15 2015 *)
RecurrenceTable[{(n+1)^2 a[n] - 2(n+2) a[n+1] + a[n+2]==0, a[1]==2, a[2]==7}, a, {n, 25}] (* Eric W. Weisstein, Sep 27 2017 *)

a(n) = sum(k=0, n, k!*binomial(n, k)^2 );

a(n) = suminf ( k=0, binomial(n+k,n)/k! ) / ( exp(1)/n! ) /* Gottfried Helms, Nov 25 2006 */

{a(n)=n!^2*polcoeff(exp(x+x*O(x^n))*sum(m=0,n,x^m/m!^2),n)} /* Paul D. Hanna, Nov 18 2011 */

{a(n)=if(n==0,1,polcoeff(1-sum(m=0, n-1, a(m)*x^m*(1-(m+1)*x+x*O(x^n))^2), n))} /* Paul D. Hanna, Nov 27 2012 */

my(x='x+O('x^22)); Vec(serlaplace((1/(1-x))*exp(x/(1-x)))) \\ Joerg Arndt, Aug 11 2022

from math import factorial, comb
def A002720(n): return sum(factorial(k)*comb(n,k)**2 for k in range(n+1)) # Chai Wah Wu, Aug 31 2023

[factorial(n)*laguerre(n, -1) for n in (0..25)] # G. C. Greubel, Aug 11 2022

a(n) = Sum_{k=0..n} k!*C(n, k)^2.
E.g.f.: (1/(1-x))*exp(x/(1-x)). - Don Knuth, Jul 1995
D-finite with recurrence: a(n) = 2*n*a(n-1) - (n-1)^2*a(n-2).
a(n) = Sum_{k>=0} (k+n)! / ((k!)^2*exp(1)). - Robert G. Wilson v, May 02 2002 [corrected by Vaclav Kotesovec, Aug 28 2012]
a(n) = Sum_{m>=0} (-1)^m*A021009(n, m). - Philippe Deléham, Mar 10 2004
a(n) = Sum_{k=0..n} C(n, k)n!/k!. - Paul Barry, May 07 2004
a(n) = Sum_{k=0..n} P(n, k)*C(n, k); a(n) = Sum_{k=0..n} n!^2/(k!*(n-k)!^2). - Ross La Haye, Sep 20 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*Bell(k+1). - Vladeta Jovovic, Mar 18 2005
Define b(n) by b(0) = 1, b(n) = b(n-1) + (1/n) * Sum_{k=0..n-1} 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) = 0.6065306597126334236... - Alec Mihailovs, Sep 06 2005, establishing a conjecture of Franklin T. Adams-Watters
a(n) = (n!/e) * Sum_{k>=0} binomial(n+k,n)/k!. - Gottfried Helms, Nov 25 2006
Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem): a(n) = Integral_{x=0..oo} x^n*BesselI(0,2*sqrt(x))*exp(-x)/exp(1) dx, n >= 0. - Karol A. Penson and G. H. E. Duchamp (gduchamp2(AT)free.fr), Jan 09 2007
a(n) = n! * LaguerreL[n, -1].
E.g.f.: exp(x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. - Paul D. Hanna, Nov 18 2011
From Peter Bala, Oct 11 2012: (Start)
Denominators in the sequence of convergents coming from Stieltjes's continued fraction for A073003, the Euler-Gompertz constant G := Integral_{x = 0..oo} 1/(1+x)*exp(-x) dx:
G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The numerators are in A002793. (End)
G.f.: 1 = Sum_{n>=0} a(n) * x^n * (1 - (n+1)*x)^2. - Paul D. Hanna, Nov 27 2012
E.g.f.: exp(x/(1-x))/(1-x) = G(0)/(1-x) where G(k) = 1 + x/((2*k+1)*(1-x) - x*(1-x)*(2*k+1)/(x + (1-x)*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 28 2012
a(n) = Sum_{k=0..n} L(n,k)*(k+1); L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014
a(n) = n! * A160617(n)/A160618(n). - Alois P. Heinz, Jun 28 2017
0 = a(n)*(-24*a(n+2) +99*a(n+3) -78*a(n+4) +17*a(n+5) -a(n+6)) +a(n+1)*(-15*a(n+2) +84*a(n+3) -51*a(n+4) +6*a(n+5)) +a(n+2)*(-6*a(n+2) +34*a(n+3) -15*a(n+4)) +a(n+3)*(+10*a(n+3)) for all n>=0. - Michael Somos, Jul 31 2018
a(n) = Sum_{k=0..n} C(n,k)*k!*A000262(n-k). - Geoffrey Critzer, Jan 07 2023
a(n) = A000262(n+1) - n * A000262(n). - Werner Schulte, Mar 29 2024
a(n) = denominator of (1 + n/(1 + n/(1 + (n-1)/(1 + (n-1)/(1 + ... + 1/(1 + 1/(1))))))). See A000262 for the numerators. - Peter Bala, Feb 11 2025

2nd description from R. H. Hardin, Nov 1997

3rd description from Wouter Meeussen, Jun 01 1998

A277373 a(n) = Sum_{k=0..n} binomial(n,n-k)n^(n-k)n!/(n-k)!.

Original entry on oeis.org

1, 2, 14, 168, 2840, 61870, 1649232, 51988748, 1891712384, 78031713690, 3598075308800, 183396819358192, 10239159335648256, 621414669926828102, 40733145577028065280, 2867932866586451980500, 215859025837098699948032, 17295664826665032427023922, 1469838791737283957748596736

Offset: 0

Peter Luschny, Oct 12 2016

Limit_{n -> infinity} (LaguerreL(n,-n)/BesselI(0,2*n))^(1/n) = exp(-2 + 1/phi) * phi^2 = 0.657347578792874..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 12 2016
For m > 0, n!*LaguerreL(n, -m*n) ~ sqrt(1/2 + (m+2)/(2*sqrt(m*(m+4)))) * (2+m+sqrt(m*(m+4)))^n * exp(n*(sqrt(m*(m+4))-m-2)/2) * n^n / 2^n. - Vaclav Kotesovec, Oct 14 2016
For m > 4, (-1)^n * n! * LaguerreL(n, m*n) ~ sqrt(1/2 + (m-2)/(2*sqrt(m*(m-4)))) * exp((m - 2 - sqrt(m*(m-4)))*n/2) * ((m - 2 + sqrt(m*(m-4)))/2)^n * n^n. - Vaclav Kotesovec, Feb 20 2020

Alois P. Heinz, Table of n, a(n) for n = 0..356
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind
Wikipedia, Laguerre polynomials

Cf. A002720 (n!L(n,-1)), A087912 (n!L(n,-2)), A277382 (n!L(n,-3)), A277372 (n!L(n,-n)-n^n), A277423 (n!L(n,n)), A144084 (polynomials).
Cf. A277391 (n!L(n,-2*n)), A277392 (n!L(n,-3*n)), A277418 (n!L(n,-4*n)), A277419 (n!L(n,-5*n)), A277420 (n!L(n,-6*n)), A277421 (n!L(n,-7*n)), A277422 (n!L(n,-8*n)).
Main diagonal of A289192.

[(&+[Binomial(n, n-k)*Binomial(n, k)*n^(n-k)*Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 16 2018

A277373 := n -> n!*LaguerreL(n, -n): seq(simplify(A277373(n)), n=0..18);
# second Maple program:
a:= n-> n! * add(binomial(n, i)*n^i/i!, i=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 27 2017

Table[n!*LaguerreL[n, -n], {n, 0, 30}] (* G. C. Greubel, May 16 2018 *)

a(n) = sum(k=0,n, binomial(n,n-k)*n^(n-k)*n!/(n-k)!) \\ Charles R Greathouse IV, Feb 07 2017

a(n) = n!*pollaguerre(n, 0, -n); \\ Michel Marcus, Feb 05 2021

@cached_function
def L(n, x):
    if n == 0: return 1
    if n == 1: return 1 - x
    return (L(n-1,x) * (2*n-1-x) - L(n-2,x)*(n-1))/n
A277373 = lambda n: factorial(n)*L(n, -n)
print([A277373(n) for n in (0..20)])

a(n) = p(n,n) where p(n,x) = Sum_{k=0..n} binomial(n,n-k)*x^(n-k)*n!/(n-k)!. The coefficients of these polynomials are in A144084 (sorted by falling powers).
a(n) = n!*LaguerreL(n, -n).
a(n) = (-1)^n*KummerU(-n, 1, -n).
a(n) = n^n*hypergeom([-n, -n], [], 1/n) for n>=1.
a(n) ~ n^n * phi^(2*n+1) * exp(n/phi-n) / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 12 2016
a(n) = n! * [x^n] exp(n*x/(1-x))/(1-x). - Alois P. Heinz, Jun 28 2017
a(n) = n!^2 * [x^n] exp(x) * BesselI(0,2*sqrt(n*x)). - Ilya Gutkovskiy, Jun 19 2022

A087912 Exponential generating function is exp(2*x/(1-x))/(1-x).

Original entry on oeis.org

1, 3, 14, 86, 648, 5752, 58576, 671568, 8546432, 119401856, 1815177984, 29808908032, 525586164736, 9898343691264, 198227905206272, 4204989697906688, 94163381359509504, 2219240984918720512, 54898699229094412288, 1422015190821016633344, 38484192401958599131136

Offset: 0

Vladeta Jovovic, Oct 18 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
OEIS Wiki, Exponential generating functions.

Cf. A002720, A274246, A277372.
Column k=2 of A289192.
Cf. A160615, A160616.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 16 2018

a := proc(n) option remember: if n<1 then 1 else (2*n+1)*a(n-1) - (n-1)^2*a(n-2) fi end: 'a(n)'$n=0..17; # Zerinvary Lajos, Sep 26 2006; corrected by M. F. Hasler, Sep 30 2012

Table[n! SeriesCoefficient[E^(2*x/(1-x))/(1-x), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, May 10 2013 *)
Table[n!*LaguerreL[n, -2], {n, 0, 30}] (* G. C. Greubel, May 16 2018 *)

A087912(n)={n!^2*polcoeff(exp(x+x*O(x^n))*sum(m=0,n,2^m*x^m/m!^2) ,n)} \\ Paul D. Hanna, Nov 18 2011

x='x+O('x^66); Vec(serlaplace(exp(2*x/(1-x))/(1-x))) \\ Joerg Arndt, May 10 2013

E.g.f.: exp(2*x/(1-x))/(1-x). - M. F. Hasler, Sep 30 2012
a(n) = n!*LaguerreL(n, -2).
a(n) = Sum_{k=0..n} 2^k*(n-k)!*binomial(n, k)^2.
E.g.f.: exp(x) * Sum_{n>=0} 2^n*x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. [Paul D. Hanna, Nov 18 2011]
a(n) ~ n^(n+1/4)*exp(2*sqrt(2*n)-n-1)*2^(-3/4). - Vaclav Kotesovec, Sep 29 2012
Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(2*n))) = exp(-1). - Vaclav Kotesovec, Oct 12 2016
a(n) = n! * A160615(n)/A160616(n). - Alois P. Heinz, Jun 28 2017
D-finite with recurrence: a(n) +(-2*n-1)*a(n-1) +(n-1)^2*a(n-2)=0. - R. J. Mathar, Feb 21 2020

Several minor edits by M. F. Hasler, Sep 30 2012

A277382 a(n) = n!*LaguerreL(n, -3).

Original entry on oeis.org

1, 4, 23, 168, 1473, 14988, 173007, 2228544, 31636449, 490102164, 8219695239, 148262469336, 2860241078817, 58736954622492, 1278727896354687, 29406849577341552, 712119108949808193, 18108134430393657636, 482306685868464422391, 13425231879291031821576

Offset: 0

Vaclav Kotesovec, Oct 12 2016

nonn

For m > 0, n!*LaguerreL(n, -m) ~ exp(2*sqrt(m*n) - n - m/2) * n^(n + 1/4) / (sqrt(2)*m^(1/4)) * (1 + (3+24*m+4*m^2)/(48*sqrt(m*n))).

Alois P. Heinz, Table of n, a(n) for n = 0..438
W. Van Assche, Erratum to "Weighted zero distribution for polynomials orthogonal on an infinite interval", SIAM J. Math. Anal., 32 (2001), 1169-1170.
Oskar Perron, Über das Verhalten einer ausgearteten hypergeometrischen Reihe bei unbegrenztem Wachstum eines Parameters, Journal für die reine und angewandte Mathematik (1921), vol. 151, p. 63-78.
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A002720, A087912, A277372.
Column k=3 of A289192.
Cf. A160613, A160614.

[Factorial(n)*((&+[Binomial(n,k)*(3^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 09 2018

Table[n!*LaguerreL[n, -3], {n, 0, 20}]
CoefficientList[Series[E^(3*x/(1-x))/(1-x), {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[Binomial[n, k]^2 * 3^k * (n-k)!, {k,0,n}], {n, 0, 20}]

for(n=0,30, print1(n!*(sum(k=0,n, binomial(n,k)*(3^k/k!))), ", ")) \\ G. C. Greubel, May 09 2018

E.g.f.: exp(3*x/(1-x))/(1-x).
a(n) = Sum_{k=0..n} 3^k*(n-k)!*binomial(n, k)^2.
a(n) ~ exp(2*sqrt(3*n)-n-3/2) * n^(n+1/4) / (sqrt(2) * 3^(1/4)) * (1 + 37/(16*sqrt(3*n))).
D-finite with recurrence a(n) = 2*(n+1)*a(n-1) - (n-1)^2*a(n-2).
Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(3*n))) = exp(-3/2).
a(n) = n! * A160613(n)/A160614(n). - Alois P. Heinz, Jun 28 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 3^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A289147 Number of (n+1) X (n+1) binary matrices M with at most one 1 in each of the first n rows and each of the first n columns and M[n+1,n+1] = 0.

Original entry on oeis.org

1, 5, 34, 286, 2840, 32344, 414160, 5876336, 91356544, 1542401920, 28075364096, 547643910400, 11389266525184, 251428006132736, 5869482147358720, 144413021660821504, 3733822274973040640, 101181690628832198656, 2867011297057247002624, 84764595415605494743040

Offset: 0

Alois P. Heinz, Jun 26 2017

nonn

Number of marriage patterns between a labeled set X of n women and a labeled set Y of n men (all heterosexual): some couples can be formed where one partner is from X and the other from Y, some members of X and Y marry external (unlabeled) partners, and some do not marry.

			a(1) = 5:
[0 0]  [1 0]  [0 1]  [0 0]  [0 1]
[0 0]  [0 0]  [0 0]  [1 0]  [1 0] .
.
a(2) = 34:
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]
[0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]
.
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]
[0 0 1]  [0 1 0]  [0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]
[1 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]
.
[0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]
[0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]
[1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]
.
[0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]
[0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]
[1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [1 0 0]
.
[0 1 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]
[1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 1 0]
[0 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]  [0 0 0]  .

Alois P. Heinz, Table of n, a(n) for n = 0..437
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=4 of A289192.

Cf.: A000142, A000165, A000302, A002720, A025167, A084771, A087912, A102773, A160611, A160612, A277382.

a:= proc(n) option remember; `if`(n<2, 4*n+1,
      (2*n+3)*a(n-1)-(n-1)^2*a(n-2))
    end:
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n-> n! * add(binomial(n, i)*4^i/i!, i=0..n):
seq(a(n), n=0..25);
# third Maple program:
a:= n-> n!* simplify(LaguerreL(n, -4), 'LaguerreL'):
seq(a(n), n=0..25);

Table[n! LaguerreL[n, -4], {n, 0, 30}] (* Indranil Ghosh, Jul 06 2017 *)

from mpmath import *
mp.dps=150
l=chop(taylor(lambda x:exp(4*x/(1-x))/(1-x), 0, 31))
print([int(fac(i)*l[i]) for i in range(len(l))]) # Indranil Ghosh, Jul 06 2017
# or #
from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -4))
print([a(n) for n in range(31)]) # Indranil Ghosh, Jul 06 2017

E.g.f.: exp(4*x/(1-x))/(1-x).
a(n) = Sum_{i=0..n} i! * (2^(n-i)*binomial(n,i))^2.
a(n) = Sum_{i=0..n} (n-i)! * 4^i * binomial(n,i)^2.
a(n) = n! * Sum_{i=0..n} 4^i/i! * binomial(n,i).
a(n) = (2*n+3)*a(n-1)-(n-1)^2*a(n-2) for n>=2, a(n) = 4*n+1 for n<2.
a(n) = n! * Laguerre(n,-4) = n! * A160611(n)/A160612(n).
a(n) ~ exp(-2 + 4*sqrt(n) - n) * n^(n + 1/4) / 2 * (1 + 163/(96*sqrt(n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 4^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A289211 a(n) = n! * Laguerre(n,-5).

Original entry on oeis.org

1, 6, 47, 446, 4929, 61870, 866695, 13373190, 224995745, 4094022230, 80031878175, 1671426609550, 37116087808225, 872797202471550, 21656891639499575, 565266064058561750, 15476777687220818625, 443409439715399299750, 13263588837009155407375

Offset: 0

Alois P. Heinz, Jun 28 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..435
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=5 of A289192.

Cf. A160609, A160610.

[(Factorial(n)*(&+[Binomial(n,k)*(5^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 09 2018

a:= n-> n! * add(binomial(n, i)*5^i/i!, i=0..n):
seq(a(n), n=0..20);

[Table[n!*LaguerreL[n,-5], {n,0,50}]] (* G. C. Greubel, May 09 2018 *)

for(n=0,30, print1(n!*sum(k=0,n, binomial(n,k)*(5^k/k!)), ", ")) \\ G. C. Greubel, May 09 2018

a(n) = n!*pollaguerre(n, 0, -5); \\ Michel Marcus, Feb 05 2021

E.g.f.: exp(5*x/(1-x))/(1-x).
a(n) = n! * Sum_{i=0..n} 5^i/i! * binomial(n,i).
a(n) = n! * A160609(n)/A160610(n).
a(n) ~ exp(-5/2 + 2*sqrt(5*n) - n) * n^(n + 1/4) / (sqrt(2)*5^(1/4)) * (1 + 223/(48*sqrt(5*n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 5^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A289212 a(n) = n! * Laguerre(n,-6).

Original entry on oeis.org

1, 7, 62, 654, 7944, 108696, 1649232, 27422352, 495057024, 9631281024, 200682406656, 4455296877312, 104921038236672, 2610989435003904, 68430995893131264, 1883330926998829056, 54286270223002140672, 1635031821385383247872, 51347572582353094508544

Offset: 0

Alois P. Heinz, Jun 28 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..434
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=6 of A289192.

Cf. A160607, A160608.

a:= n-> n! * add(binomial(n, i)*6^i/i!, i=0..n):
seq(a(n), n=0..20);

Table[n!*LaguerreL[n, -6], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)

my(x = 'x + O('x^30)); Vec(serlaplace(exp(6*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017

a(n) = n!*pollaguerre(n, 0, -6); \\ Michel Marcus, Feb 05 2021

from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -6))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

E.g.f.: exp(6*x/(1-x))/(1-x).
a(n) = n! * Sum_{i=0..n} 6^i/i! * binomial(n,i).
a(n) = n! * A160607(n)/A160608(n).
a(n) ~ exp(-3 + 2*sqrt(6*n) - n) * n^(n + 1/4) / (2^(3/4)*3^(1/4)) * (1 + 97/(16*sqrt(6*n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 6^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A289214 a(n) = n! * Laguerre(n,-8).

Original entry on oeis.org

1, 9, 98, 1238, 17688, 280888, 4894672, 92676144, 1891712384, 41361536384, 963532779264, 23806296960256, 621407739476992, 17078348640463872, 492724294444623872, 14883442560920164352, 469591293625846038528, 15443081743064125505536, 528340631093887891603456

Offset: 0

Alois P. Heinz, Jun 28 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..431
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=8 of A289192.

Cf. A160603, A160604.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(8*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 11 2018

a:= n-> n! * add(binomial(n, i)*8^i/i!, i=0..n):
seq(a(n), n=0..20);

Table[n!*LaguerreL[n, -8], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)

my(x = 'x + O('x^30)); Vec(serlaplace(exp(8*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017

a(n) = n!*pollaguerre(n, 0, -8); \\ Michel Marcus, Feb 05 2021

from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -8))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

E.g.f.: exp(8*x/(1-x))/(1-x).
a(n) = n! * Sum_{i=0..n} 8^i/i! * binomial(n,i).
a(n) = n! * A160603(n)/A160604(n).
a(n) ~ exp(-4 + 4*sqrt(2*n) - n) * n^(n + 1/4) / 2^(5/4) * (1 + 451/(96*sqrt(2*n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 8^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A289215 a(n) = n! * Laguerre(n,-9).

Original entry on oeis.org

1, 10, 119, 1626, 24945, 422994, 7836255, 157169826, 3388099329, 78031713690, 1910451937671, 49510386761130, 1353167691897969, 38878205830928226, 1170930069982659375, 36875214316479123954, 1211549306913066598785, 41445016025330141416746

Offset: 0

Alois P. Heinz, Jun 28 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..430
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=9 of A289192.

Cf. A160601, A160602.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(9*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 11 2018

a:= n-> n! * add(binomial(n, i)*9^i/i!, i=0..n):
seq(a(n), n=0..20);

Table[n!*LaguerreL[n, -9], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)

my(x = 'x + O('x^30)); Vec(serlaplace(exp(9*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017

a(n) = n!*pollaguerre(n, 0, -9); \\ Michel Marcus, Feb 05 2021

from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -9))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

E.g.f.: exp(9*x/(1-x))/(1-x).
a(n) = n! * Sum_{i=0..n} 9^i/i! * binomial(n,i).
a(n) = n! * A160601(n)/A160602(n).
a(n) ~ exp(-9/2 + 6*sqrt(n) - n) * n^(n + 1/4) / sqrt(6) * (1 + 181/(48*sqrt(n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 9^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A289216 a(n) = n! * Laguerre(n,-10).

Original entry on oeis.org

1, 11, 142, 2086, 34184, 616120, 12083920, 255749840, 5801633920, 140276126080, 3598075308800, 97512721964800, 2782552712473600, 83347512973644800, 2613606571616819200, 85594543750221568000, 2921314815145299968000, 103704333851191177216000

Offset: 0

Alois P. Heinz, Jun 28 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..429
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=10 of A289192.

Cf. A160587, A160589.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(10*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 11 2018

a:= n-> n! * add(binomial(n, i)*10^i/i!, i=0..n):
seq(a(n), n=0..20);

Table[n!*LaguerreL[n, -10], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)

my(x = 'x + O('x^30)); Vec(serlaplace(exp(10*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017

a(n) = n!*pollaguerre(n, 0, -10); \\ Michel Marcus, Feb 05 2021

from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -10))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

E.g.f.: exp(10*x/(1-x))/(1-x).
a(n) = n! * Sum_{i=0..n} 10^i/i! * binomial(n,i).
a(n) = n! * A160587(n)/A160589(n).
a(n) ~ exp(-5 + 2*sqrt(10*n) - n) * n^(n + 1/4) / (2^(3/4)*5^(1/4)) * (1 + 643/(48*sqrt(10*n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 10^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

cp's OEIS Frontend

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.

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A277373 a(n) = Sum_{k=0..n} binomial(n,n-k)*n^(n-k)*n!/(n-k)!.

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A087912 Exponential generating function is exp(2*x/(1-x))/(1-x).

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A277382 a(n) = n!*LaguerreL(n, -3).

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A289147 Number of (n+1) X (n+1) binary matrices M with at most one 1 in each of the first n rows and each of the first n columns and M[n+1,n+1] = 0.

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A289211 a(n) = n! * Laguerre(n,-5).

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A277373 a(n) = Sum_{k=0..n} binomial(n,n-k)n^(n-k)n!/(n-k)!.