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A331320 a(n) = [x^n] ((x + 1)*(2*x - 1)*(2*x^2 - 1))/(2*x^2 + 2*x - 1)^2.

%I A331320 #12 Jun 14 2025 18:55:03
%S A331320 1,3,8,26,80,244,736,2200,6528,19248,56448,164768,478976,1387328,
%T A331320 4005376,11530624,33107968,94839552,271091712,773380608,2202374144,
%U A331320 6261404672,17774206976,50384312320,142636515328,403306786816,1139055820800,3213593911296,9057375289344
%N A331320 a(n) = [x^n] ((x + 1)*(2*x - 1)*(2*x^2 - 1))/(2*x^2 + 2*x - 1)^2.
%H A331320 Colin Barker, <a href="/A331320/b331320.txt">Table of n, a(n) for n = 0..1000</a>
%H A331320 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-8,-4).
%F A331320 a(n) = Sum_{k=0..n} A322942(n,k)*(k+1).
%F A331320 a(n) = (12*(n - 3)*a(n-3) + (14*n - 6)*a(n-2) + (70 - 4*n)*a(n-1))/(n + 19).
%F A331320 Let h(k) = (1+k)*exp((1+k)*x)*(3*x+12-4*k)/18 then
%F A331320 a(n) = n!*[x^n](h(sqrt(3)) + h(-sqrt(3)) + 1).
%F A331320 From _Colin Barker_, Jan 14 2020: (Start)
%F A331320 a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4) for n>4.
%F A331320 a(n) = (-8*sqrt(3)*((1-sqrt(3))^n - (1+sqrt(3))^n) + 3*((1-sqrt(3))^n + (1+sqrt(3))^n)*n) / 18 for n>0.
%F A331320 (End)
%p A331320 a := proc(n) option remember; if n < 3 then return [1, 3, 8][n+1] fi;
%p A331320 (12*(n - 3)*a(n-3) + (14*n - 6)*a(n-2) + (70 - 4*n)*a(n-1))/(n+19) end:
%p A331320 seq(a(n), n=0..28);
%p A331320 # Alternative:
%p A331320 gf := ((x + 1)*(2*x - 1)*(2*x^2 - 1))/(2*x^2 + 2*x - 1)^2:
%p A331320 ser := series(gf, x, 32): seq(coeff(ser, x, n), n=0..28);
%t A331320 LinearRecurrence[{4,0,-8,-4},{1,3,8,26,80},40] (* _Harvey P. Dale_, Jun 14 2025 *)
%o A331320 (PARI) Vec((1 + x)*(1 - 2*x)*(1 - 2*x^2) / (1 - 2*x - 2*x^2)^2 + O(x^30)) \\ _Colin Barker_, Jan 14 2020
%Y A331320 Cf. A322942 (Jacobsthal triangle), A331319, A331321.
%K A331320 nonn,easy
%O A331320 0,2
%A A331320 _Peter Luschny_, Jan 14 2020