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A005649
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Expansion of e.g.f. (2 - e^x)^(-2).
(Formerly M1866)
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77
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1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548, 54667412, 862440068, 14857100084, 277474957988, 5584100659412, 120462266974148, 2772968936479604, 67843210855558628, 1757952715142990612, 48093560991292628228, 1385244691781856307124
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OFFSET
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0,2
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COMMENTS
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Exponential self-convolution of numbers of preferential arrangements.
Number of compatible bipartitional relations on a set of cardinality n. - Ralf Stephan, Apr 27 2003
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
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
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:
(1) (1,2) (1,2,1)
(2,1) (1,2,3)
(1,3,2)
(2,1,2)
(2,1,3)
(2,3,1)
(3,1,2)
(3,2,1)
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:
{{1}} {{1},{2}} {{1,3},{2}}
{{2},{1}} {{2},{1,3}}
{{1},{2},{3}}
{{1},{3},{2}}
{{2},{1},{3}}
{{2},{3},{1}}
{{3},{1},{2}}
{{3},{2},{1}}
(End)
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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E.g.f.: 1/(2-exp(x))^2.
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
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
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
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
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
a(n) = Sum_{k=0..n} Stirling2(n,k)*(k + 1)!. - Ilya Gutkovskiy, Jul 25 2018
a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
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)
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MAPLE
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b:= proc(n, m) option remember;
`if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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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 *)
a[n_] := (-1)^n (PolyLog[-n-1, 2] - PolyLog[-n, 2])/4; Array[f, 20, 0] (* Vladimir Reshetnikov, Jan 23 2011 *)
Range[0, 19]! CoefficientList[Series[(2 - Exp@ x)^-2, {x, 0, 19}], x] (* Robert G. Wilson v, Jan 23 2011 *)
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 *)
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n], FreeQ[Differences[#], 0]&]], {n, 0, 6}] (* Gus Wiseman, Jun 10 2020 *)
With[{nn=20}, CoefficientList[Series[1/(2-E^x)^2, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 02 2021 *)
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PROG
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(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst(1/(1-y)^2, y, exp(x+x*O(x^n))-1), n))
(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)
(Maxima) t(n):=sum(stirling2(n, k)*k!, k, 0, n);
makelist(sum(binomial(n, k)*t(k)*t(n-k), k, 0, n), n, 0, 20);
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CROSSREFS
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Anti-run compositions are counted by A003242.
A triangle counting maximal anti-runs of compositions is A106356.
Anti-runs of standard compositions are counted by A333381.
Adjacent unequal pairs in standard compositions are counted by A333382.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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