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A020556 Number of oriented multigraphs on n labeled arcs (without loops). 20
1, 1, 7, 87, 1657, 43833, 1515903, 65766991, 3473600465, 218310229201, 16035686850327, 1356791248984295, 130660110400259849, 14177605780945123273, 1718558016836289502159, 230999008481288064430879, 34208659263890939390952225, 5549763869122023099520756513 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Generalized Bell numbers: a(n) = sum(A078739(n,k),k=2..2*n),n>=1.

Let B_{m}(x) = Sum_{j>=0} exp(j!/(j-m)!*x-1)/j!  then

a(n) = n! [x^n] taylor(B_{2}(x)), where [x^n] denotes the coefficient of x^n in the Taylor series for B_{2}(x). a(n) is row 2 of the square array representation of A090210. - Peter Luschny, Mar 27 2011

Also the number of set partitions of {1,2,..,2n+1} such that the block |n+1| is a part but no block |m| with m < n+1. - Peter Luschny, Apr 03 2011

REFERENCES

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..288

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, Phys. Lett. A 309 (2003) 198-205.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

P. Codara, O. M. D'Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013

G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.

Peter Luschny, Set partitions

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

M. Riedel, Set partitions of unique elements from an n-by-m matrix where elements from the same row may not be in the same partition

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

FORMULA

a(n) = Sum( (k+2)!^n /(k+2)!*(k!^n)*exp(1)), k = 0 .. infinity, n>=1.

a(n) = (sum(((k*(k-1))^n)/k!, k=2..infinity)/exp(1), n>=1. a(0) := 1. (From eq.(26) with r=2 of the Schork reference.)

E.g.f.: (sum((exp(k*(k-1)*x))/k!, k=2..infinity)+2)/exp(1) (from top of p. 4656 of the Schork reference).

a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*Bell(2*n-k). - Vladeta Jovovic, May 02 2004

EXAMPLE

Example: For n = 2 the a(2) = 7 are the number of set partitions of 5 such that the block |3| is a part but no block |m| with m < 3: 3|1245, 3|4|125, 3|5|124, 3|12|45, 3|14|25, 3|15|24, 3|4|5|12. - Peter Luschny, Apr 05 2011

MAPLE

A020556 := proc(n) local k;

add((-1)^(n+k)*binomial(n, k)*combinat[bell](n+k), k=0..n) end:

seq(A020556(n), n=0..17); # Peter Luschny, Mar 27 2011

# Uses floating point arithmetic, increase working precision for large n.

A020556 := proc(n) local r, s, i;

if n=0 then 1 else r := [seq(3, i=1..n-1)]; s := [seq(1, i=1..n-1)];

exp(-x)*2^(n-1)*hypergeom(r, s, x); round(evalf(subs(x=1, %), 99)) fi end:

seq(A020556(n), n=0..15); # Peter Luschny, Mar 30 2011

T := proc(n, k) option remember;

  if n = 1 then 1

elif n = k then T(n-1, 1) - T(n-1, n-1)

else T(n-1, k) + T(n, k+1) fi end:

A020556 := n -> T(2*n+1, n+1);

seq(A020556(n), n = 0..99); # Peter Luschny, Apr 03 2011

MATHEMATICA

f[n_] := f[n] = Sum[(k + 2)!^n/((k + 2)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 16}]

(* Second program: *)

a[n_] := Sum[(-1)^k*Binomial[n, k]*BellB[2n-k], {k, 0, n}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 11 2017, after Vladeta Jovovic *)

CROSSREFS

Cf. A020554, A014500, A020558, A090210.

Sequence in context: A199027 A279845 A231447 * A303061 A007803 A034219

Adjacent sequences:  A020553 A020554 A020555 * A020557 A020558 A020559

KEYWORD

nonn

AUTHOR

Gilbert Labelle (gilbert(AT)lacim.uqam.ca) and Simon Plouffe

EXTENSIONS

Edited by Robert G. Wilson v, Apr 30 2002

STATUS

approved

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Last modified September 18 11:42 EDT 2018. Contains 315130 sequences. (Running on oeis4.)