|
| |
|
|
A000247
|
|
2^n-n-2.
(Formerly M2836 N1141)
|
|
8
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 2,2
|
|
|
COMMENTS
| Ways of placing n+1 labeled balls into 2 indistinguishable boxes with at least 2 balls in each box.
2^a(n) = integer values 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)=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 (callan(AT)stat.wisc.edu), Dec 08 2004
A107907(a(n)) = A000225(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 28 2005
Subsequence of A158581; A000120(a(n)) > 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 16 2009]
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
|
|
|
REFERENCES
| Amer. Math. Monthly, Vol. 101 (No. 8, Oct 1994), p. 776.
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).
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=2..300
L. E. Jeffery, Unit-primitive matrices
T. Mansour, Restricted permutations by patterns of type 2-1.
Mathoverflow, Face numbers for tropical Grassmannian G'_2,7 simplicial complex?
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Alex Vinokur, Fibonacci-like polynomials produced by m-ary Huffman codes for absolutely ordered sequences, arXiv:cs/0411002
Index to sequences with linear recurrences with constant coefficients, signature (4,-5,2).
|
|
|
FORMULA
| E.g.f.: (exp(x)-1-x)*(exp(x)-1). G.f.: (3-2*x)*x^3/((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 (qntmpkt(AT)yahoo.com), 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) [From Harvey P. Dale, Aug 23 2011]
a(n) = sum(2<=k<(n+1)/2, C(n+1,k)) + if(n odd, C(n+1,(n+1)/2)/2, 0).
|
|
|
EXAMPLE
| a(3)=4!/(2!*2!*2!)=3
|
|
|
MAPLE
| with(combinat):a:=n->sum(sum(binomial(j, k), j=2..n), k=1..n): seq(a(n), n=1..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 10 2007
A000247:=(-3+2*z)/((2*z-1)*(z-1)**2); [S. Plouffe in his 1992 dissertation.]
a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=n+2*a[n-1]+1 od: seq(a[n], n=1..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008
restart:with (combinat):a:=n->add(stirling2(j, 2), j=3..n): seq(a(n), n=2..25); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 01 2009]
|
|
|
MATHEMATICA
| lst={}; s=1; Do[s+=(s-n); AppendTo[lst, Abs[s]], {n, 2, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 10 2008]
Table[Sum[Binomial[n, i - 1], {i, 3, n}], {n, 2, 41}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]
LinearRecurrence[{4, -5, 2}, {0, 3, 10}, 50] (* From Harvey P. Dale, Aug 23 2011 *)
|
|
|
PROG
| (Other) sage: [gaussian_binomial(n, 1, 2)-(n+1) for n in xrange(2, 32)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009]
|
|
|
CROSSREFS
| Cf. A000478 (3 boxes), A058844 (4 boxes).
Sequence in context: A162607 A047667 A192963 * A097763 A034506 A067988
Adjacent sequences: A000244 A000245 A000246 * A000248 A000249 A000250
|
|
|
KEYWORD
| nonn,easy,nice,changed
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Additional comments from Michael Steyer (msteyer(AT)osram.de), Dec 02 2000. More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000
I recently changed the beginning of this sequence so the formulae etc. may need to be adjusted. - N. J. A. Sloane, Jan 24 2006
Formulas and comments adjusted for offset by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 10 2011
|
| |
|
|
