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A000247
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a(n) = 2^n - n - 2.
(Formerly M2836 N1141)
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21
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0, 3, 10, 25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, 65518, 131053, 262124, 524267, 1048554, 2097129, 4194280, 8388583, 16777190, 33554405, 67108836, 134217699, 268435426, 536870881, 1073741792, 2147483615
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OFFSET
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2,2
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COMMENTS
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Ways of placing n+1 labeled balls into 2 indistinguishable boxes with at least 2 balls in each box.
2^a(n) is an integer of the form 1/(2 - Sum_{i=1..m} i/2^i). - Benoit Cloitre, Oct 25 2002
Number of permutations avoiding 13-2 that contain the pattern 23-1 exactly twice.
Cost of ternary maximum height Huffman tree with N internal nodes (non-leaves) for minimizing absolutely ordered sequences of size n = 2N + 1. - Alex Vinokur (alexvn(AT)barak-online.net), Nov 02 2004
a(n) is the number of Dyck n-paths whose third upstep initiates the last long ascent, n >= 1. A long ascent is one consisting of 2 or more upsteps. For example, a(3)=3 counts UUDuUDDD, UDUDuUDD, UUDDuUDD (third upstep in small type). - David Callan, Dec 08 2004
Number of vertices of the tropical Grassmannian simplicial complex G(2,n), related to phylogenetic trees. - Tom Copeland, Oct 03 2011
(Conjecture) Let a(2)=0. For n > 2, let N = 2*n + 1. For each n, define the n X n tridiagonal unit-primitive matrix (see [Jeffery]) A_{N,1}=[0,1,0,...,0; 1,0,1,0,...,0; 0,1,0,1,0,...,0; ...; 0,...,0,1,0,1; 0,...,0,1,1] associated with N. Define the n-dimensional column vectors V_N = [v_1,v_2,...,v_n]^T = [A_{N,1}]^n*[1,1,...,1]^T, where [.]^T denotes matrix transpose and [1,...,1] is the n-dimensional unit vector. Let (v_k)_N denote the k-th element of V_N, k in {1,...,n}. Then a(n) = (v_(n-2))_N. - L. Edson Jeffery, Jan 20 2012
For n>2, a(n-2) is the number of connected induced (non-null) subgraphs of the n-centipede graph. - Giovanni Resta, May 04 2017
a(n) is the number of maximal boundary strata of the moduli space of stable rational curves with n+1 marked points. The closures of the maximal boundary strata are called the irreducible boundary divisors of the moduli space; see Cavalieri Section 2.1. - Harry Richman, Aug 13 2024
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.: (exp(x)-1-x)*(exp(x)-1).
G.f.: x^3*(3-2*x)/((1-2*x)*(1-x)^2).
a(n) = 2*a(n-1) + n + 3 = a(n-1) + 2^(n-1) - 1 = A000295(n) - 1 = A000295(n+1) - 2^(n+1).
Starting (3, 10, 25, 56, ...) = binomial transform of [3, 7, 8, 8, 8, ...]. - Gary W. Adamson, Nov 07 2007
a(2)=0, a(3)=3, a(4)=10, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Harvey P. Dale, Aug 23 2011
a(n) = (Sum_{k=2..floor(n/2)} binomial(n+1,k)) + if(n odd, binomial(n+1,(n+1)/2)/2, 0).
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EXAMPLE
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a(3) = 4!/(2!*2!*2!) = 3.
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MAPLE
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MATHEMATICA
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LinearRecurrence[{4, -5, 2}, {0, 3, 10}, 40] (* Harvey P. Dale, Aug 23 2011 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000
I recently changed the beginning of this sequence so the formulas etc. may need to be adjusted. - N. J. A. Sloane, Jan 24 2006
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STATUS
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approved
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