A000486 One half of the number of permutations of [n] such that the differences have 4 runs with the same signs.
16, 150, 926, 4788, 22548, 100530, 433162, 1825296, 7577120, 31130190, 126969558, 515183724, 2082553132, 8395437930, 33776903714, 135691891272, 544517772984, 2183315948550, 8748985781230, 35043081823140, 140313684667076
Offset: 5
Examples
a(5)=16 because the permutations of [5] with four sign runs are 13254, 14253, 14352, 15342, 15243, 21435, 21534, 23154, 24153, 25143, 31425, 31524, 32415, 32514, 41325, 42315 and their reversals.
References
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #13
- F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
- 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
- Vincenzo Librandi, Table of n, a(n) for n = 5..1000
- Index entries for linear recurrences with constant coefficients, signature (13,-67,175,-244,172,-48).
Programs
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Mathematica
CoefficientList[Series[2 (24 x^2 - 29 x + 8)/((x - 1)^2 (2 x - 1)^2 (3 x - 1) (4 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 13 2013 *) -
PARI
a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -48,172,-244,175,-67,13]^(n-5)*[16;150;926;4788;22548;100530])[1,1] \\ Charles R Greathouse IV, Jun 23 2020
Formula
Limit_{n->infinity} 8*a(n)/4^n = 1. - Philippe Deléham, Feb 22 2004
G.f.: 2*x^5*(24*x^2-29*x+8) / ((x-1)^2*(2*x-1)^2*(3*x-1)*(4*x-1)). - Colin Barker, Dec 21 2012
Extensions
Edited by Emeric Deutsch, Feb 18 2004