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 A080599 Expansion of e.g.f.: 2/(2-2*x-x^2). 16
 1, 1, 3, 12, 66, 450, 3690, 35280, 385560, 4740120, 64751400, 972972000, 15949256400, 283232149200, 5416632421200, 110988861984000, 2425817682288000, 56333385828720000, 1385151050307024000, 35950878932544576000, 982196278209226080000, 28175806418228108640000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of ordered partitions of {1,..,n} with at most 2 elements per block. - Bob Proctor, Apr 18 2005 In other words, number of preferential arrangements of n things (see A000670) in which each clump has size 1 or 2. - N. J. A. Sloane, Apr 13 2014 Recurrences (of the hypergeometric type of the Jovovic formula) mean: multiplying the sequence vector from the left with the associated matrix of the recurrence coefficients (here: an infinite lower triangular matrix with the natural numbers in the main diagonal and the triangular series in the subdiagonal) recovers the sequence up to an index shift. In that sense, this sequence here and many other sequences of the OEIS are eigensequences. - Gary W. Adamson, Feb 14 2011 Number of intervals in the weak (Bruhat) order of S_n that are Boolean algebras. - Richard Stanley, May 09 2011 a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+2*x)*d/dx. Cf. A000085, A005442 and A052585. - Peter Bala, Dec 07 2011 From Gus Wiseman, Jul 04 2020: (Start) Also the number of (1,1,1)-avoiding or cubefree sequences of length n covering an initial interval of positive integers. For example, the a(0) = 1 through a(3) = 12 sequences are:   ()  (1)  (11)  (112)            (12)  (121)            (21)  (122)                  (123)                  (132)                  (211)                  (212)                  (213)                  (221)                  (231)                  (312)                  (321) (End) LINKS Alois P. Heinz, Table of n, a(n) for n = 0..200 L. Gellert, R. Sanyal, On degree sequences of undirected, directed, and bidirected graphs, arXiv preprint arXiv:1512.08448 [math.CO], 2015. Dixy Msapato, Counting the number of tau-exceptional sequences over Nakayama algebras, arXiv:2002.12194 [math.RT], 2020. Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007. FORMULA a(n) = n*a(n-1)+(n*(n-1)/2)*a(n-2). - Vladeta Jovovic, Aug 22 2003 E.g.f.: 1/(1-x-x^2/2). - Richard Stanley, May 09 2011 a(n) ~ n!*((1+sqrt(3))/2)^(n+1)/sqrt(3). - Vaclav Kotesovec, Oct 13 2012 a(n) = n!*((1+sqrt(3))^(n+1) - (1-sqrt(3))^(n+1))/(2^(n+1)*sqrt(3)). - Vladimir Reshetnikov, Oct 31 2015 a(n) = A090932(n) * A002530(n+1). - Robert Israel, Nov 01 2015 EXAMPLE From Gus Wiseman, Jul 04 2020: (Start) The a(0) = 1 through a(3) = 12 ordered set partitions with block sizes <= 2 are:   {}  {{1}}  {{1,2}}    {{1},{2,3}}              {{1},{2}}  {{1,2},{3}}              {{2},{1}}  {{1,3},{2}}                         {{2},{1,3}}                         {{2,3},{1}}                         {{3},{1,2}}                         {{1},{2},{3}}                         {{1},{3},{2}}                         {{2},{1},{3}}                         {{2},{3},{1}}                         {{3},{1},{2}}                         {{3},{2},{1}} (End) MAPLE a:= n-> n! *(Matrix([[1, 1], [1/2, 0]])^n)[1, 1]: seq(a(n), n=0..40);  # Alois P. Heinz, Jun 01 2009 a:= gfun:-rectoproc({a(n) = n*a(n-1)+(n*(n-1)/2)*a(n-2), a(0)=1, a(1)=1}, a(n), remember): seq(a(n), n=0..40); # Robert Israel, Nov 01 2015 MATHEMATICA Table[n!*SeriesCoefficient[-2/(-2+2*x+x^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2012 *) Round@Table[n! ((1+Sqrt[3])^(n+1) - (1-Sqrt[3])^(n+1))/(2^(n+1) Sqrt[3]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *) PROG (PARI) Vec(serlaplace((2/(2-2*x-x^2) + O(x^30)))) \\ Michel Marcus, Nov 02 2015 CROSSREFS Cf. A000085, A000670, A002530, A005442, A052585, A052611, A090932. Column k=2 of A276921. Cubefree numbers are A004709. (1,1)-avoiding patterns are A000142. (1,1,1)-avoiding compositions are A232432. (1,1,1)-matching patterns are A335508. (1,1,1)-avoiding permutations of prime indices are A335511. (1,1,1)-avoiding compositions are ranked by A335513. (1,1,1,1)-avoiding patterns are A189886. Cf. A102726, A106356, A232464, A333755, A335455, A335456. Sequence in context: A199746 A293302 A248871 * A120575 A009362 A123227 Adjacent sequences:  A080596 A080597 A080598 * A080600 A080601 A080602 KEYWORD nonn,eigen AUTHOR Detlef Pauly (dettodet(AT)yahoo.de), Feb 24 2003 STATUS approved

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Last modified June 23 08:56 EDT 2021. Contains 345395 sequences. (Running on oeis4.)