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A257198 Number of permutations of length n having exactly one descent such that the first element of the permutation is an odd number.

%I A257198 #44 Sep 08 2022 08:46:12
%S A257198 0,0,2,6,16,36,78,162,332,672,1354,2718,5448,10908,21830,43674,87364,
%T A257198 174744,349506,699030,1398080,2796180,5592382,11184786,22369596,
%U A257198 44739216,89478458,178956942,357913912,715827852,1431655734,2863311498,5726623028
%N A257198 Number of permutations of length n having exactly one descent such that the first element of the permutation is an odd number.
%H A257198 Colin Barker, <a href="/A257198/b257198.txt">Table of n, a(n) for n = 1..1000</a>
%H A257198 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,2).
%F A257198 a(n) = 2*floor((2*2^n-3*n-1)/6).
%F A257198 a(n) = 2*A178420(n-1).
%F A257198 a(n) = A000295(n)-A000975(n-1).
%F A257198 From _Colin Barker_, Apr 19 2015: (Start)
%F A257198 a(n) = (-3-(-1)^n+2^(2+n)-6*n)/6.
%F A257198 a(n) = 3*a(n-1)-a(n-2)-3*a(n-3)+2*a(n-4).
%F A257198 G.f.: -2*x^3 / ((x-1)^2*(x+1)*(2*x-1)).
%F A257198 (End)
%e A257198 a(3)=2: (1 3 2, 3 1 2).
%e A257198 a(4)=6: (1 2 4 3, 1 3 2 4, 1 4 2 3, 1 3 4 2, 3 1 2 4, 3 4 1 2).
%t A257198 Table[2 Floor[(2 2^n - 3 n - 1) / 6], {n, 50}] (* _Vincenzo Librandi_, Apr 18 2015 *)
%o A257198 (Magma) [2*Floor((2*2^n-3*n-1)/6): n in [1..40]]; // _Vincenzo Librandi_, Apr 18 2015
%o A257198 (PARI) concat([0,0], Vec(-2*x^3/((x-1)^2*(x+1)*(2*x-1)) + O(x^100))) \\ _Colin Barker_, Apr 19 2015
%o A257198 (PARI) a(n)=(2<<n-3*n-1)\6*2 \\ _Charles R Greathouse IV_, Apr 21 2015
%Y A257198 Cf. A178420, A000295, A000975, A167030 (first differences).
%K A257198 nonn,easy
%O A257198 1,3
%A A257198 _Ran Pan_, Apr 18 2015