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%I M1866 #109 Nov 19 2023 08:24:54
%S 1,2,8,44,308,2612,25988,296564,3816548,54667412,862440068,
%T 14857100084,277474957988,5584100659412,120462266974148,
%U 2772968936479604,67843210855558628,1757952715142990612,48093560991292628228,1385244691781856307124
%N Expansion of e.g.f. (2 - e^x)^(-2).
%C Exponential self-convolution of numbers of preferential arrangements.
%C Number of compatible bipartitional relations on a set of cardinality n. - _Ralf Stephan_, Apr 27 2003
%C Stirling transform of A000142, shifted left one place: 1, 2, 6, 24, 120, 720, ... - _Philippe Deléham_, May 17 2005; corrected by _Ilya Gutkovskiy_, Jul 25 2018
%C With an extra 1 at the beginning, coefficients of the formal (divergent) series expansion at infinity of Sum_{k>=0} 1/binomial(x,k) = 1+1/x+2/x^2+8/x^3+... Also Sum_{k>=0} k!/x^k Product_{i=1..k-1} 1/(1-i/x) yields a generating function in 1/x. - _Roland Bacher_, Nov 21 2000
%C Stirling-Bernoulli transform of A001057: 1, -1, 2, -2, 3, -3, 4, ... - _Philippe Deléham_, May 27 2015
%C a(n) is the total number of open sets summed over all chain topologies that can be placed on an n-set. A chain topology is a topology whose open sets can be totally ordered by inclusion. - _Geoffrey Critzer_, Apr 06 2017
%C From _Gus Wiseman_, Jun 10 2020: (Start)
%C Also the number of length n + 1 sequences covering an initial interval of positive integers with no adjacent equal parts (anti-runs). For example, the a(0) = 1 through a(2) = 8 anti-runs are:
%C (1) (1,2) (1,2,1)
%C (2,1) (1,2,3)
%C (1,3,2)
%C (2,1,2)
%C (2,1,3)
%C (2,3,1)
%C (3,1,2)
%C (3,2,1)
%C Also the number of ordered set partitions of {1,...,n + 1} with no two successive vertices in the same block. For example, the a(0) = 1 through a(2) = 8 ordered set partitions are:
%C {{1}} {{1},{2}} {{1,3},{2}}
%C {{2},{1}} {{2},{1,3}}
%C {{1},{2},{3}}
%C {{1},{3},{2}}
%C {{2},{1},{3}}
%C {{2},{3},{1}}
%C {{3},{1},{2}}
%C {{3},{2},{1}}
%C (End)
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A005649/b005649.txt">Table of n, a(n) for n = 0..423</a> (first 101 terms from T. D. Noe)
%H José A. Adell, Beáta Bényi, Venkat Murali, and Sithembele Nkonkobe, <a href="https://doi.org/10.22108/toc.2022.130037.1894">Generalized Barred Preferential Arrangements</a>, Transactions on Combinatorics (2022).
%H Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i2p55">Barred Preferential Arrangements</a>, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
%H D. Foata, and C. Krattenthaler, <a href="http://www.mat.univie.ac.at/~kratt/artikel/graphmaj.html">Graphical Major Indices, II</a>, Séminaire Lotharingien de Combinatoire, B34k, 16 pp., 1995.
%H D. Foata and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/9406220">The Graphical Major Index</a>, arXiv:math/9406220 [math.CO], 1994.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=154">Encyclopedia of Combinatorial Structures 154</a>
%H Augustine O. Munagi, <a href="https://www.fq.math.ca/Papers1/55-5/Munagi.pdf">Two Applications of the Bijection on Fibonacci Set Partitions</a>, Fibonacci Quart. 55 (2017), no. 5, 144-148.
%F E.g.f.: 1/(2-exp(x))^2.
%F a(n) = (A000670(n) + A000670(n+1)) / 2. - _Philippe Deléham_, May 16 2005
%F a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000670 and A052841. - _Peter Bala_, Nov 25 2011
%F E.g.f.: 1/(2-exp(x))^2 = 1/(G(0) + 4), G(k) = 1-4/((2^k)-x*(4^k)/((2^k)*x-(2*k+2)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Dec 15 2011
%F O.g.f.: Sum_{n>=0} (2*n)!/n! * x^n / Product_{k=1..n} (1 + (n+k)*x). - _Paul D. Hanna_, Jan 03 2013
%F G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (k+1)/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 15 2013
%F G.f.: 1/G(0) where G(k) = 1 - x*(k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction ). - _Sergei N. Gladkovskii_, Mar 23 2013
%F a(n) = Sum_{k = 0..n} A163626(n,k) * A001057(k+1). - _Philippe Deléham_, May 27 2015
%F a(n) ~ n! * n / (4 * (log(2))^(n+2)). - _Vaclav Kotesovec_, Jul 01 2018
%F a(n) = Sum_{k=0..n} Stirling2(n,k)*(k + 1)!. - _Ilya Gutkovskiy_, Jul 25 2018
%F From _Seiichi Manyama_, Nov 19 2023: (Start)
%F a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
%F a(0) = 1; a(n) = 2*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)
%p b:= proc(n, m) option remember;
%p `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..23); # _Alois P. Heinz_, Aug 03 2021
%t f[n_] := Sum[(i + j)^n/2^(2 + i + j), {i, 0, Infinity}, {j, 0, Infinity}]; Array[f, 20, 0] (* _Vladimir Reshetnikov_, Dec 31 2008 *)
%t a[n_] := (-1)^n (PolyLog[-n-1, 2] - PolyLog[-n, 2])/4; Array[f, 20, 0] (* _Vladimir Reshetnikov_, Jan 23 2011 *)
%t Range[0, 19]! CoefficientList[Series[(2 - Exp@ x)^-2, {x, 0, 19}], x] (* _Robert G. Wilson v_, Jan 23 2011 *)
%t nn = 19; Range[0, nn]! CoefficientList[Series[1 + D[u^2 (Exp[z] - 1)/(1 - u (Exp[z] - 1)), u] /. u -> 1, {z, 0, nn}], z] (* _Geoffrey Critzer_, Apr 06 2017 *)
%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t Table[Length[Select[Join@@Permutations/@allnorm[n],FreeQ[Differences[#],0]&]],{n,0,6}] (* _Gus Wiseman_, Jun 10 2020 *)
%t With[{nn=20},CoefficientList[Series[1/(2-E^x)^2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 02 2021 *)
%o (PARI) a(n)=if(n<0,0,n!*polcoeff(subst(1/(1-y)^2,y,exp(x+x*O(x^n))-1),n))
%o (PARI) a(n)=polcoeff(sum(m=0, n,(2*m)!/m!*x^m/prod(k=1, m,1+(m+k)*x+x*O(x^n))), n)
%o for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jan 03 2013
%o (Maxima) t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o makelist(sum(binomial(n,k)*t(k)*t(n-k),k,0,n),n,0,20);
%o \\ _Emanuele Munarini_, Oct 02 2012
%Y Cf. A000670.
%Y 2*A083410(n)=a(n), if n>0.
%Y Pairwise sums of A052841 and also of A089677.
%Y Anti-run compositions are counted by A003242.
%Y A triangle counting maximal anti-runs of compositions is A106356.
%Y Anti-runs of standard compositions are counted by A333381.
%Y Adjacent unequal pairs in standard compositions are counted by A333382.
%Y Cf. A000110, A008277, A055932, A124762, A238279, A238424, A317081.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
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