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%I #12 Mar 09 2023 08:42:00
%S 0,0,0,0,0,4,18,55,151,399,1061,2792,7252,18572,47051,118291,295718,
%T 735776,1823183,4501151,11077398,27187053,66567357,162655518,
%U 396728446,966109074,2349342360,5705883591,13842565488,33549053111,81238182806,196560818348
%N Number of self-inverse permutations p on [n] where the maximal displacement of an element equals 4.
%H Joerg Arndt and Alois P. Heinz, <a href="/A238915/b238915.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_23">Index entries for linear recurrences with constant coefficients</a>, signature (3, 0, -3, 1, 2, -7, -16, -7, -21, -12, -9, 1, 16, 7, 11, 0, 3, 2, 1, 1, -2, -1, -1).
%F G.f.: (x^12 -x^9 -x^8 +2*x^7 -x^6 +3*x^5 -4*x^4 -2*x^3 +x^2 +6*x +4)*x^5 / ((x+1)*(x^6 -x^5 +x^4 -3*x^3 +3*x^2 -3*x+1) *(x^16 +x^15 +2*x^14 +x^13 +x^12 +2*x^11 +x^10 +3*x^9 -4*x^8 -5*x^7 -9*x^6 -6*x^5 -x^4 -x^3 -2*x^2 -x+1)).
%e a(5) = 4: 52341, 52431, 53241, 54321.
%e a(6) = 18: 163452, 163542, 164352, 165432, 361452, 361542, 463152, 465132, 523416, 523614, 524316, 526413, 532416, 532614, 543216, 546213, 563412, 564312.
%e a(7) = 55: 1274563, 1274653, 1275463, ..., 5637124, 5643127, 5674123.
%p gf:= (x^12 -x^9 -x^8 +2*x^7 -x^6 +3*x^5 -4*x^4 -2*x^3 +x^2 +6*x +4)*x^5 /
%p ((x+1)*(x^6 -x^5 +x^4 -3*x^3 +3*x^2 -3*x+1) *(x^16 +x^15 +2*x^14 +x^13 +x^12 +2*x^11 +x^10 +3*x^9 -4*x^8 -5*x^7 -9*x^6 -6*x^5 -x^4 -x^3 -2*x^2 -x+1)):
%p a:= n-> coeff(series(gf, x, n+1), x, n):
%p seq(a(n), n=0..40);
%t CoefficientList[Series[(x^12 - x^9 - x^8 + 2 x^7 - x^6 + 3 x^5 - 4 x^4 - 2 x^3 + x^2 + 6 x + 4) x^5/((x + 1) (x^6 - x^5 + x^4 - 3 x^3 + 3 x^2 - 3 x + 1) (x^16 + x^15 + 2 x^14 + x^13 + x^12 + 2 x^11 + x^10 + 3 x^9 - 4 x^8 - 5 x^7 - 9 x^6 - 6 x^5 - x^4 - x^3 - 2 x^2 - x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 09 2014 *)
%t LinearRecurrence[{3,0,-3,1,2,-7,-16,-7,-21,-12,-9,1,16,7,11,0,3,2,1,1,-2,-1,-1},{0,0,0,0,0,4,18,55,151,399,1061,2792,7252,18572,47051,118291,295718,735776,1823183,4501151,11077398,27187053,66567357},40] (* _Harvey P. Dale_, Mar 09 2023 *)
%Y Column k=4 of A238889.
%K nonn
%O 0,6
%A _Joerg Arndt_ and _Alois P. Heinz_, Mar 07 2014
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