A069223 -id:A069223 - 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

A078741 Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).

Original entry on oeis.org

1, 6, 18, 9, 1, 36, 540, 1242, 882, 243, 27, 1, 216, 13608, 94284, 186876, 149580, 56808, 11025, 1107, 54, 1, 1296, 330480, 6148872, 28245672, 49658508, 41392620, 18428400, 4691412, 706833, 63375, 3285, 90, 1, 7776, 7954848, 380841264, 3762380016, 13062960720

Offset: 1

N. J. A. Sloane, Dec 21 2002

The sequence of row lengths for this array is [1,4,7,10,..]= A016777(n-1), n>=1.
The g.f. for the k-th column, (with leading zeros and k>=3) is G(k,x)= x^ceiling(k/3)*P(k,x)/product(1-fallfac(p,3)*x,p=3..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A089517(k,m)*x^m,m=0..kmax(k)), k>=3, with kmax(k) := A004523(k-3)= floor(2*(k-3)/3)= [0,0,1,2,2,3,4,4,5,...]. For the recurrence of the G(k,x) see A089517. Wolfdieter Lang, Dec 01 2003
For the computation of the k-th column sequence see A090219.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_3 (the disjoint union of n copies of the complete graph K_3). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k |   3     4     5      6      7     8     9   10  11  12
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
  1 |   1
  2 |   6    18     9      1
  3 |  36   540  1242    882    243    27     1
  4 | 216 13608 94284 186876 149580 56808 11025 1107  54   1
...
Graph coloring interpretation of T(2,3) = 6:
The graph 2K_3 is 2 copies of K_3, the complete graph on 3 vertices:
    o b      o e
   / \      / \
  o---o    o---o
  a   c    d   f
The six 3-colorings of 2K_3 are ad|be|cf, ad|bf|ce, ae|bd|cf, ae|bf|cd, af|bd|ce, and af|be|cd. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027,2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Codara, O. M. D’Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers arXiv:1308.1700v1 [cs.DM]
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
W. Lang, First 6 rows.

Row sums give A069223. Cf. A078739.
The column sequences (without leading zeros) are A000400 (powers of 6), 18*A089507, 9*A089518, A089519, etc.
A089504, A069223 (row sums), A090212 (alternating row sums).
A078740, A090214.

a[n_, k_] := (-1)^k*Sum[(-1)^p*((p-2)*(p-1)*p)^n*Binomial[k, p], {p, 3, k}]/k!; Table[a[n, k], {n, 1, 6}, {k, 3, 3*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

a(n, k) = (((-1)^k)/k!)*Sum_{p = 3..k} (-1)^p* binomial(k, p)*fallfac(p, 3)^n, with fallfac(p, 3) := A008279(p, 3) = p*(p-1)*(p-2); 3 <= k <= 3*n, n >= 1, else 0. From eq.(19) with r = 3 of the Blasiak et al. reference.
E^n = Sum_{k = 3..3*n} a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^3d^3/dx^3.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k >= 0} (k*(k-1)*(k-2))^n*x^k/k!. - Peter Bala, Aug 15 2013

A090210 Triangle of certain generalized Bell numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 7, 1, 1, 15, 87, 34, 1, 1, 52, 1657, 2971, 209, 1, 1, 203, 43833, 513559, 163121, 1546, 1, 1, 877, 1515903, 149670844, 326922081, 12962661, 13327, 1, 1, 4140, 65766991, 66653198353, 1346634725665, 363303011071, 1395857215, 130922, 1, 1

Offset: 1

Wolfdieter Lang, Dec 01 2003

Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) and
S(n,k) = k! [x^k] taylor(B_{n}(x)), where [x^k] denotes the
coefficient of x^k in the Taylor series for B_{n}(x).
Then S(n,k) (n>0, k>=0) is the square array representation of the triangle.
To illustrate the cross-references of T(n,k) when written as a square array.
n\k         A000012,A000012,A002720,A069948,A182925,A182924,...
0: A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1: A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
2: A020556: 1, 1, 7, 87, 1657, 43833, 1515903, ...
3: A069223: 1, 1, 34, 2971, 513559, 149670844, ...
4: A071379: 1, 1, 209, 163121, 326922081, ...
5: A090209: 1, 1, 1546, 12962661, 363303011071,...
6:  ...     1, 1, 13327, 1395857215, 637056434385865,...
Note that the sequence T(0,k) is not included in the data.
- Peter Luschny, Mar 27 2011

			Triangle begins:
1;
1, 1;
2, 1, 1;
5, 7, 1, 1;
15, 87, 34, 1, 1;
52, 1657, 2971, 209, 1, 1;
203, 43833, 513559, 163121, 1546, 1, 1;

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
W. Lang, First 8 rows.
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp,Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).

Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182924, A182925, A182933.

T(n,n) gives A070227.

A090210_AsSquareArray := proc(n,k) local r,s,i;
if k=0 then 1 else r := [seq(n+1,i=1..k-1)]; s := [seq(1,i=1..k-1)];
exp(-x)*n!^(k-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(lprint(seq(A090210_AsSquareArray(n,k),k=0..6)),n=0..6);
# Peter Luschny, Mar 30 2011

t[n_, k_] := t[n, k] = Sum[(n+j)!^(k-1)/(j!^k*E), {j, 0, Infinity}]; t[_, 0] = 1;
Flatten[ Table[ t[n-k+1, k], {n, 0, 8}, {k, n, 0, -1}]][[1 ;; 43]] (* Jean-François Alcover, Jun 17 2011 *)

a(n, m) = Bell(m;n-(m-1)), n>= m-1 >=0, with Bell(m;k) := Sum_{p=m..m*k} S2(m;k, p), where S2(m;k, p) := (((-1)^p)/p!) * Sum_{r=m..p} ((-1)^r)*binomial(p, r)*fallfac(r, m)^k; with fallfac(n, m) := A008279(n, m) (falling factorials) and m<=p<=k*m, k>=1, m=1, 2, ..., else 0. From eqs.(6) with r=s->m and eq.(19) with S_{r, r}(n, k)-> S2(r;n, k) of the Blasiak et al. reference. [Corrected by Sean A. Irvine, Jun 03 2024]
a(n, m) = (Sum_{k>=m} fallfac(k, m)^(n-(m-1)))/exp(1), n>=m-1>=0, else 0. From eq.(26) with r->m of the Schork reference which is rewritten eq.(11) of the original Blasiak et al. reference.
E.g.f. m-th column (no leading zeros): (Sum_{k>=m} exp(fallfac(k, m)*x)/k!) + A000522(m)/m!)/exp(1). Rewritten from the top of p. 4656 of the Schork reference.

A071379 a(n) = (1/e) * Sum_{k>=0} ((k+4)!/k!)^(n-1)/k!.

Original entry on oeis.org

1, 1, 209, 163121, 326922081, 1346634725665, 9939316337679281, 119802044788535500753, 2205421644124274191535553, 58945667435045762187763602753, 2198513228897522394476415669503377, 110833342180980170285766876408530089329

Offset: 0

Karol A. Penson, May 22 2002

nonn

This is a Dobinski-type summation formula.
Terms quickly become gigantic: a(15) = 9142140479823239889945170786704021785456107245847570873873. a(n) appears in the process of ordering the n-th power of a product of fourth power of boson creation and fourth power of boson annihilation operators.
From Peter Luschny, Mar 27 2011: (Start)
Let B_{m}(x) = Sum_{j>=0} (exp(j!/(j-m)!*x-1)/j!) then a(n) = n! [x^n] Taylor(B_{4}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{4}(x).
a(n) is row 4 of the square array representation of A090210. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A000110, A020556 and A069223, when k+4 is replaced by k+1, k+2 or k+3 respectively.

Cf. A090210.

A071379 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(5,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*24^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A071379(n),n=0..10); # Peter Luschny, Mar 30 2011

a[n_] := Sum[FactorialPower[k, 4]^n/k!, {k, 4, Infinity}]/E; a[0] = 1; Array[a, 12, 0] (* Jean-François Alcover, Sep 01 2016 *)

default(realprecision, 500); for(n=0, 20, print1(if(n==0, 1, round(exp(-1)*sum(k=0, 500, ((k+4)!/k!)^(n-1)/k!))), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = (1/e)*Sum_{k>=4} fallfac(k, 4)^n / k!, n >= 1, with fallfac(n, m) := A008279(n, m) (falling factorials). (From eq.(26) with r=4 of the Schork reference.)

E.g.f. with a(0) := 1: (1/e)*(Sum_{k>=4} e^(fallfac(k, 4)*x)/k! + 8/3). From top of p. 4656 with r=4 of the Schork reference.

a(0)=1 prepended by Alois P. Heinz, Aug 01 2016

If it is proved that A283153 and A071379 are the same, then the entries should be merged and A283153 recycled. - N. J. A. Sloane, Mar 06 2017

A069948 a(n) = 1/exp(1) * Sum_{k>=0} (k+n)!^2 / k!^3.

Original entry on oeis.org

1, 5, 87, 2971, 163121, 12962661, 1395857215, 194634226067, 33990369362241, 7247035915622821, 1848636684656077991, 555005864462114884875, 193458213840943964983537, 77399534126148191747554181, 35196002960227350045891984591, 18037244723394790042393195636291

Offset: 0

Robert G. Wilson v, May 02 2002

nonn

From Peter Luschny, Mar 27 2011: (Start)
Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) then a(n) = 3! [x^3] taylor(B_{n}(x)), where [x^3] denotes the coefficient of x^3 in the Taylor series for B_{n}(x).
a(n) is column 3 of the square array representation of A090210. (End)

G. C. Greubel, Table of n, a(n) for n = 0..240
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)

Cf. A000110, A020556, A069223, A090210.

A069948 := proc(n) exp(-x)*n!^2*hypergeom([n+1,n+1],[1,1],x);
round(evalf(subs(x=1,%),99)) end:
seq(A069948(n),n=0..13); # Peter Luschny, Mar 30 2011
# second Maple program:
a:= n-> sum((k+n)!^2/k!^3, k=0..infinity)/exp(1):
seq(a(n), n=0..15);  # Alois P. Heinz, May 17 2018

f[n_] := f[n] = Sum[(k + n)!^3/((k + n)!*(k!^3)*E), {k, 0, Infinity}]; Table[ f[n], {n, 0, 13}] (* or *)
Table[n!^2*HypergeometricPFQ[{n + 1, n + 1}, {1, 1}, 1]/Exp[1], {n, 0, 13}] (* Robert G. Wilson v, Jan 11 2007 *)

{default(realprecision, 200)}; for(n=0,30, print1(round(exp(-1)*(n!)^2*sum(k=0,500, binomial(n+k, k)^2/k!)), ", ")) \\ G. C. Greubel, May 17 2018

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*2*BesselK(0,2*sqrt(x))*hypergeom([],[1,1],x)/exp(1), x=0..infinity), n=0,1... Special values of the hypergeometric function of type 2F2: a(n)=exp(-1)*GAMMA(n+1)^2*hypergeom([n+1, n+1], [1, 1], 1). - Karol A. Penson and G. H. E. Duchamp (gduchamp2(AT)free.fr), Jan 09 2007
Recurrence: (8*n-7)*a(n) = (24*n^3 + 3*n^2 - 26*n + 4)*a(n-1) - (n-1)^2*(24*n^3 - 85*n^2 + 66*n + 13)*a(n-2) + (n-1)^2*(8*n+1)*(n-2)^4*a(n-3). - Vaclav Kotesovec, Jul 30 2013
a(n) ~ n^(2*n+1/3)*exp(n^(1/3) + 3*n^(2/3) - 2*n - 2/3)/sqrt(3) * (1 + 41/(54*n^(1/3)) + 13769/(29160*n^(2/3))). - Vaclav Kotesovec, Jul 30 2013

More terms from Robert G. Wilson v, Jan 11 2007

A090209 Generalized Bell numbers (from (5,5)-Stirling2 array A090216).

Original entry on oeis.org

1, 1, 1546, 12962661, 363303011071, 25571928251231076, 3789505947767235111051, 1049433111253356296672432821, 498382374325731085522315594481036, 380385281554629647028734545622539438171, 443499171330317702437047276255605780991365151

Offset: 0

Wolfdieter Lang, Dec 01 2003

Contribution from Peter Luschny, Mar 27 2011: (Start) Let B_{m}(x) = sum_{j>=0}(exp(j!/(j-m)!*x-1)/j!) then a(n) = n! [x^n] taylor(B_{5}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{5}(x).

a(n) is row 5 of the square array representation of A090210. (End)

M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

G. C. Greubel, Table of n, a(n) for n = 0..115
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-ph/0402027, Phys. Lett. A 309 (3-4) (2003) 198-205
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).

Cf. (Generalized) Bell numbers from (m,m)-Stirling2 array: A000110 (m=1), A020556 (m=2), A069223 (m=3), A071379 (m=4). Triangle A090210.

A071379 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(6,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*5!^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A071379(n),n=0..8); # Peter Luschny, Mar 30 2011

fallfac[n_, k_] := Pochhammer[n-k+1, k]; a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*fallfac[p, 5]^n, {p, 5, k}]; a[0] = 1; a[n_] := Sum[a[n, k], {k, 5, 5*n}];  Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Mar 05 2014 *)

a(n) = Sum_{k=5..5*n} A090216(n, k), n>=1. a(0) := 1.
a(n) = Sum_{k >=5} (fallfac(k, 5)^n)/k!/exp(1), n>=1, a(0) := 1. From eq.(26) with r=5 of the Schork reference.
E.g.f. with a(0) := 1: (sum((exp(fallfac(k, 5)*x))/k!, k=5..infinity)+ A000522(4)/4!)/exp(1). From the top of p. 4656 with r=5 of the Schork reference.

If it is proved that A283154 and A090209 are the same, then the entries should be merged and A283154 recycled. - N. J. A. Sloane, Mar 06 2017

A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 5, 27, 13, 1, 1, 15, 409, 778, 73, 1, 1, 52, 9089, 104149, 37553, 501, 1, 1, 203, 272947, 25053583, 57184313, 2688546, 4051, 1, 1, 877, 10515147, 9566642254, 192052025697, 56410245661, 265141267, 37633, 1

Offset: 0

Peter Luschny, Mar 29 2011

These numbers are related to the generalized Bell numbers based on the falling factorial powers (A090210).
The square array starts for n>=0, k>=0:
n\k=0,1,..  A000012,A000262,A182934,...
A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
A094577: 1, 3, 27, 409, 9089, 272947, 10515147, ...
A182932: 1, 13, 778, 104149, 25053583, 9566642254, ...
       : 1, 73, 37553, 57184313, 192052025697, ...
       : 1, 501, 2688546, 56410245661, ...
 ....  : 1, 4051, 265141267, 89501806774945, ...

Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182925, A182924, A182933.

A182933_AsSquareArray := proc(n,k) local r,s,i;
r := [seq(n+1,i=1..k)]; s := [seq(1,i=1..k-1),2];
exp(-x)*n!^k*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) end:
seq(lprint(seq(A182933_AsSquareArray(n,k),k=0..6)),n=0..6);

a[n_, k_] := Exp[-1]*n!^k*HypergeometricPFQ[ Table[n+1, {k}], Append[ Table[1, {k-1}], 2], 1.]; Table[ a[n-k, k] // Round , {n, 0, 8}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Let r_k = [n+1,...,n+1] (k occurrences of n+1), s_k = [1,...,1,2] (k-1 occurrences of 1) and F_k the generalized hypergeometric function of type k_F_k, then a(n,k) = exp(-1)*n!^k*F_k(r_k, s_k | 1).

Let B_{n}(x) = sum_{j>=0}(exp((j+n-1)!/(j-1)!*x-1)/j!) then a(n,k) = k! [x^k] series(B_{n}(x)), where [x^k] denotes the coefficient of x^k in the Taylor series for B_{n}(x).

A283153 Number of set partitions of unique elements from an n X 4 matrix where elements from the same row may not be in the same partition.

Original entry on oeis.org

1, 209, 163121, 326922081, 1346634725665, 9939316337679281, 119802044788535500753, 2205421644124274191535553, 58945667435045762187763602753, 2198513228897522394476415669503377, 110833342180980170285766876408530089329, 7356710448423295420590529054176924329802529, 628972339934967292421997567343442748145219556449

Offset: 1

Marko Riedel, Mar 01 2017

nonn

Apparently a duplicate of A071379? - R. J. Mathar, Mar 06 2017

Cf. A283154, A283155, A071379.

Table[(4!^n) * Sum[Binomial[p,4]^n/p! * Sum[(-1)^k/k!,{k,0,4n-p}],{p,1,4n}],{n,1,50}] (* Indranil Ghosh, Mar 04 2017 *)

a(n) = (4!^n) * sum(p=1, 4*n, binomial(p,4)^n/p! * sum(k=0, 4*n-p, (-1)^k/k!)); \\ Indranil Ghosh, Mar 04 2017

a(n) = m!^n * Sum_{p=1..n*m} (binomial(p,m)^n/p!) * Sum_{k=0..n*m-p} (-1)^k/k! with m=4.

If it is proved that A283153 and A071379 are the same, then the entries should be merged and A283153 recycled. - N. J. A. Sloane, Mar 06 2017

A078738 Generalized Bell numbers B_{3,2}(n).

Original entry on oeis.org

1, 13, 355, 16333, 1121881, 106708921, 13354028563, 2118817455385, 414426460442833, 97746679844312581, 27311169061720393411, 8908525371578726747173, 3350963996380181114090665, 1438463413778071631322236593, 698374517715612292764726380851

Offset: 1

N. J. A. Sloane, Dec 21 2002

Vaclav Kotesovec, Table of n, a(n) for n = 1..240
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

B_{1, 1} = A000110, B_{2, 1} = A000262, B_{3, 1} = A020556 and B_{3, 3} = A069223. Row sums of A078740.

Alternating row sums A090437.

a[n_] := (n+1)*n!^2*Sum[(-1)^k*HypergeometricPFQ[{2-k, n+1, n+2}, {2, 3}, 1]/(2*(k-2)!), {k, 2, 2n}]; Array[a, 13] (* Jean-François Alcover, Sep 01 2015 *)
Table[Sum[(n + k)!*(n + k + 1)!/(k!*(k + 1)!*(k + 2)!), {k, 0, Infinity}]/E, {n, 1, 20}] (* Vaclav Kotesovec, Jul 27 2018 *)

nmax = 20; p = floor(3*nmax*log(nmax)); default(realprecision, p);
for(n=1, nmax, print1(round(exp(-1)*suminf(k=0, (n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))), ", ")) \\ G. C. Greubel and Vaclav Kotesovec, Jul 28 2018

a(n) = Sum_{k=2..2*n} A078740(n, k) = Sum_{k=1..infinity} (1/k!)*Product_{j=1..n}(fallfac(k+(j-1)*(3-2), 2))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=3, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
a(n) = Sum_{k>=0} ((n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))/exp(1), n>=1. From eq.(40) of the Blasiak et al. reference. [corrected by Vaclav Kotesovec, Jul 27 2018]
E.g.f. for a(n)/n! with a(0)=(exp(1)-1)/exp(1) added: Sum_{k>=0} (hypergeom([k+2, k+1], [1], z)/(k+2)!)/exp(1). From eq. (41) of the Blasiak et al. reference.

Edited by Wolfdieter Lang, Dec 23 2003

A157280 a(n) arises in the normal ordering of n-th power of the operator (d/dx)(x(d/dx))^4.

Original entry on oeis.org

1, 52, 43833, 149670844, 1346634725665, 25571928251231076, 893591647147188285577, 52327970757667659912764908, 4796836032234830356783078467969

Offset: 1

Karol A. Penson, Feb 26 2009

nonn

this sequence generates the fifth terms of the following sequences:
a(2)=52=A000110(5), a(3)=43833=A020556(5), a(4)=149670844=A069223(5),
a(5)=1346634725665=A071379(5),a(6)=25571928251231076=A070227(5)

nMax = 8; kMax = 50; seq0 = {}; seq = {1}; While[seq != seq0, seq0 = seq; seq = (1/E Sum[HypergeometricPFQ[{k+1, k+1, k+1, k+1}, {1, 1, 1, 1}, x]/k!, {k, 0, kMax}] + O[x]^(nMax+1) // CoefficientList[#, x]&) Range[0, nMax]!^5 // Round; kMax += 10; Print[kMax]]; A157280 = seq (* Jean-François Alcover, Nov 07 2016 *)

Sequence defined through the following hypergeometric-type generating function, in Maple notation:
exp(-1)*sum(hypergeom([k+1,k+1,k+1,k+1],[1,1,1,1],x)/k!,k=0..infinity)=sum(a(n)*x^n/(n!)^5,n=0..infinity),
which is itself an infinite sum of hypergeometric functions.

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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A078741 Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).

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A090210 Triangle of certain generalized Bell numbers.

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A071379 a(n) = (1/e) * Sum_{k>=0} ((k+4)!/k!)^(n-1)/k!.

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A069948 a(n) = 1/exp(1) * Sum_{k>=0} (k+n)!^2 / k!^3.

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A090209 Generalized Bell numbers (from (5,5)-Stirling2 array A090216).

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