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A000486 One half of the number of permutations of [n] such that the differences have 4 runs with the same signs.

%I A000486 M5011 N2158 #33 Jan 05 2022 00:31:00
%S A000486 16,150,926,4788,22548,100530,433162,1825296,7577120,31130190,
%T A000486 126969558,515183724,2082553132,8395437930,33776903714,135691891272,
%U A000486 544517772984,2183315948550,8748985781230,35043081823140,140313684667076
%N A000486 One half of the number of permutations of [n] such that the differences have 4 runs with the same signs.
%D A000486 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #13
%D A000486 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
%D A000486 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000486 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000486 Vincenzo Librandi, <a href="/A000486/b000486.txt">Table of n, a(n) for n = 5..1000</a>
%H A000486 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (13,-67,175,-244,172,-48).
%F A000486 Limit_{n->infinity} 8*a(n)/4^n = 1. - _Philippe Deléham_, Feb 22 2004
%F A000486 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
%e A000486 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.
%t A000486 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 *)
%o A000486 (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
%Y A000486 a(n) = T(n, 4), where T(n, k) is the array defined in A008970.
%Y A000486 Equals 1/2 * A060158(n).
%K A000486 nonn,easy
%O A000486 5,1
%A A000486 _N. J. A. Sloane_
%E A000486 Edited by _Emeric Deutsch_, Feb 18 2004