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A000431 Expansion of 2*x^3/((1-2*x)^2*(1-4*x)).
(Formerly M2089 N0824)
7
0, 0, 0, 2, 16, 88, 416, 1824, 7680, 31616, 128512, 518656, 2084864, 8361984, 33497088, 134094848, 536608768, 2146926592, 8588754944, 34357248000, 137433710592, 549744803840, 2199000186880, 8796044787712, 35184271425536, 140737278640128, 562949517213696 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Number of permutations of length n with exactly one valley. Also (for n>0), the number of ways to pick two of the 2^(n-1) vertices of an n-1 cube that are not connected by an edge. - Aaron Meyerowitz, Apr 21 2014

a(n+1), n >= 1: Number of independent vertex pairs for Q_n, n >= 1: 2^(n-1) * (2^n - (n+1)) = T_(2^n - 1) - n * 2^(n-1) = L_n - E_n = A006516(n) - A001787(n), where L_n is the number of vertex pairs and E_n is the number of vertex pairs yielding edges. (Cf. A027624.) - Daniel Forgues, Feb 19 2015

REFERENCES

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261.

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 = 0..200

S. Billey, K. Burdzy and B. E. Sagan, Permutations with given peak set, arXiv preprint arXiv:1209.0693 [math.CO], 2012. - From N. J. A. Sloane, Dec 26 2012

C. J. Fewster, D. Siemssen, Enumerating Permutations by their Run Structure, arXiv preprint arXiv:1403.1723 [math.CO], 2014.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

R. G. Rieper and M. Zeleke, Valleyless Sequences, arXiv:math/0005180 [math.CO], 2000.

Index entries for linear recurrences with constant coefficients, signature (8,-20,16)

FORMULA

From Mitch Harris, Apr 02 2004: (Start)

a(n) = Sum[k A007814(k), {k, 1, 2^(n+1) - 1}].

a(n) = (4^n - n 2^(n+1))/8 for n >= 1.

(End)

a(n) = 2*A100575(n-1). - R. J. Mathar, Mar 14 2011

a(n) = 2^(n-2) * (2^(n-1) - n), n >= 1. - Daniel Forgues, Feb 24 2015

MAPLE

A000431:=-2/(4*z-1)/(-1+2*z)**2; # conjectured by Simon Plouffe in his 1992 dissertation

a:= n-> if n=0 then 0 else (Matrix([[2, 0, 0]]). Matrix(3, (i, j)-> if (i=j-1) then 1 elif j=1 then [8, -20, 16][i] else 0 fi)^(n-1))[1, 3] fi: seq(a(n), n=0..30); # Alois P. Heinz, Aug 26 2008

MATHEMATICA

nn = 30; CoefficientList[Series[2*x^3/((1 - 2*x)^2*(1 - 4*x)), {x, 0, nn}], x] (* T. D. Noe, Jun 20 2012 *)

Join[{0}, LinearRecurrence[{8, -20, 16}, {0, 0, 2}, 30]] (* Jean-François Alcover, Jan 31 2016 *)

PROG

(MAGMA) [0] cat [(4^n - n*2^(n+1))/8: n in [1..30]]; // Vincenzo Librandi, Feb 18 2015

(PARI) concat(vector(3), Vec(2*x^3/((1-2*x)^2*(1-4*x)) + O(x^40))) \\ Michel Marcus, Jan 31 2016

CROSSREFS

Cf. A000487, A000517, A027624.

Column k=1 of A008303.

Sequence in context: A071893 A220505 A069440 * A281982 A207595 A253487

Adjacent sequences:  A000428 A000429 A000430 * A000432 A000433 A000434

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 30 04:27 EDT 2017. Contains 287305 sequences.