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A020556
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Number of oriented multigraphs on n labeled arcs (without loops).
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18
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1, 1, 7, 87, 1657, 43833, 1515903, 65766991, 3473600465, 218310229201, 16035686850327, 1356791248984295, 130660110400259849, 14177605780945123273, 1718558016836289502159, 230999008481288064430879
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 3 2011
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REFERENCES
| G. Labelle, Counting enriched multigraphs..., Discrete Math., 217 (2000), 237-248.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
G. Paquin, D\'enombrement de multigraphes enrichis, M\'emoire, Math. Dept., Univ. Qu\'ebec \`a Montr\'eal, 2004.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
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LINKS
| 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.
Peter Luschny,Set partitions
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FORMULA
| Sum( (k+2)!^n /(k+2)!*(k!^n)*exp(1)), k = 0 .. infinity, n>=1.
(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 (vladeta(AT)eunet.rs), May 02 2004
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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 5 2011
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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 3 2011
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MATHEMATICA
| f[n_] := f[n] = Sum[(k + 2)!^n/((k + 2)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 16}]
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CROSSREFS
| Cf. A020554, A014500, A020558, A090210.
Sequence in context: A183613 A173812 A199027 * A007803 A034219 A034238
Adjacent sequences: A020553 A020554 A020555 * A020557 A020558 A020559
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KEYWORD
| nonn
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AUTHOR
| Gilbert Labelle (gilbert(AT)lacim.uqam.ca) and Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 30 2002
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