cp's OEIS Frontend

A020556 Number of oriented multigraphs on n labeled arcs (without loops).

%I A020556 #75 Aug 25 2022 03:38:57
%S A020556 1,1,7,87,1657,43833,1515903,65766991,3473600465,218310229201,
%T A020556 16035686850327,1356791248984295,130660110400259849,
%U A020556 14177605780945123273,1718558016836289502159,230999008481288064430879,34208659263890939390952225,5549763869122023099520756513
%N A020556 Number of oriented multigraphs on n labeled arcs (without loops).
%C A020556 Generalized Bell numbers: a(n) = Sum_{k=2..2*n} A078739(n,k), n >= 1.
%C A020556 Let B_{m}(x) = Sum_{j>=0} exp(j!/(j-m)!*x-1)/j! then
%C A020556 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
%C A020556 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
%D A020556 G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
%H A020556 Alois P. Heinz, <a href="/A020556/b020556.txt">Table of n, a(n) for n = 0..288</a>
%H A020556 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>, arXiv:quant-ph/0212072, 2002.
%H A020556 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://dx.doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem</a>, Phys. Lett. A 309 (2003) 198-205.
%H A020556 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H A020556 P. Codara, O. M. D'Antona, P. Hell, <a href="http://arxiv.org/abs/1308.1700">A simple combinatorial interpretation of certain generalized Bell and Stirling numbers</a>, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
%H A020556 G. Labelle, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00265-4">Counting enriched multigraphs according to the number of their edges (or arcs)</a>, Discrete Math., 217 (2000), 237-248.
%H A020556 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions</a>
%H A020556 G. Paquin, <a href="/A038205/a038205.pdf">Dénombrement de multigraphes enrichis</a>, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
%H A020556 M. Riedel, <a href="http://math.stackexchange.com/questions/2165093/">Set partitions of unique elements from an n-by-m matrix where elements from the same row may not be in the same partition</a>, Mathematics Stack Exchange.
%H A020556 M. Schork, <a href="http://dx.doi.org/10.1088/0305-4470/36/16/314">On the combinatorics of normal ordering bosonic operators and deforming it</a>, J. Phys. A 36 (2003) 4651-4665.
%F A020556 a(n) = e*Sum_{k>=0} ((k+2)!^n/(k+2)!)*(k!^n), n>=1.
%F A020556 a(n) = (1/e)*Sum_{k>=2} (k*(k-1))^n/k!, n >= 1. a(0) := 1. (From eq.(26) with r=2 of the Schork reference.)
%F A020556 E.g.f.: (1/e)*(2 + Sum_{k>=2} ((exp(k*(k-1)*x))/k!)) (from top of p. 4656 of the Schork reference).
%F A020556 a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*Bell(2*n-k). - _Vladeta Jovovic_, May 02 2004
%F A020556 a(n) = A095149(2n,n). - _Alois P. Heinz_, Dec 20 2018
%F A020556 a(n) = A106436(2n,n) = A182930(2n+1,n+1). - _Alois P. Heinz_, Jan 29 2019
%e A020556 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
%p A020556 A020556 := proc(n) local k;
%p A020556 add((-1)^(n+k)*binomial(n,k)*combinat[bell](n+k),k=0..n) end:
%p A020556 seq(A020556(n),n=0..17); # _Peter Luschny_, Mar 27 2011
%p A020556 # Uses floating point arithmetic, increase working precision for large n.
%p A020556 A020556 := proc(n) local r,s,i;
%p A020556 if n=0 then 1 else r := [seq(3,i=1..n-1)]; s := [seq(1,i=1..n-1)];
%p A020556 exp(-x)*2^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
%p A020556 seq(A020556(n),n=0..15); # _Peter Luschny_, Mar 30 2011
%p A020556 T := proc(n, k) option remember;
%p A020556   if n = 1 then 1
%p A020556 elif n = k then T(n-1,1) - T(n-1,n-1)
%p A020556 else T(n-1,k) + T(n, k+1) fi end:
%p A020556 A020556 := n -> T(2*n+1,n+1);
%p A020556 seq(A020556(n), n = 0..99); # _Peter Luschny_, Apr 03 2011
%t A020556 f[n_] := f[n] = Sum[(k + 2)!^n/((k + 2)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 16}]
%t A020556 (* Second program: *)
%t A020556 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_ *)
%o A020556 (PARI) a(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(k=0, n, (-1)^k*binomial(n,k)*polcoef(bell, 2*n-k))} \\ _Andrew Howroyd_, Jan 13 2020
%Y A020556 Cf. A020554, A014500, A020558, A090210, A095149, A106436, A182930.
%K A020556 nonn
%O A020556 0,3
%A A020556 Gilbert Labelle (gilbert(AT)lacim.uqam.ca) and _Simon Plouffe_
%E A020556 Edited by _Robert G. Wilson v_, Apr 30 2002