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A331319 a(n) = [x^n](x - 2*x^3)/(1 - 2*x*(x + 1))^2.

%I A331319 #18 Jan 15 2020 05:14:28
%S A331319 0,1,4,14,48,156,496,1544,4736,14352,43072,128224,379136,1114560,
%T A331319 3260160,9494656,27545600,79642880,229573632,659951104,1892478976,
%U A331319 5414755328,15461117952,44064835584,125371383808,356137570304,1010187124736,2861518086144,8095486246912
%N A331319 a(n) = [x^n](x - 2*x^3)/(1 - 2*x*(x + 1))^2.
%H A331319 Colin Barker, <a href="/A331319/b331319.txt">Table of n, a(n) for n = 0..1000</a>
%H A331319 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-8,-4).
%F A331319 a(n) = Sum_{k=0..n} A322942(n, k)*k.
%F A331319 a(n) = 2*((n^2 - n - 2)*a(n-2) + (n^2 - 2*n - 4)*a(n-1))/(n^2 - 3*n).
%F A331319 a(n) = n! [x^n] (1/9)*exp(x)*(sqrt(3)*(3*x+2)*sinh(sqrt(3)*x)+3*x*cosh(sqrt(3)*x)).
%F A331319 From _Colin Barker_, Jan 14 2020: (Start)
%F A331319 a(n) = ((1-sqrt(3))^n*(-2*sqrt(3) + 3*n) + (1+sqrt(3))^n*(2*sqrt(3) + 3*n)) / 18.
%F A331319 a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4) for n>3.
%F A331319 (End)
%p A331319 gf := (x - 2*x^3)/(1 - 2*x*(x + 1))^2: ser := series(gf, x, 32):
%p A331319 seq(coeff(ser, x, n), n=0..28);
%t A331319 LinearRecurrence[ {4, 0, -8, -4}, {0, 1, 4, 14}, 28]
%o A331319 (PARI) concat(0, Vec(x*(1 - 2*x^2) / (1 - 2*x - 2*x^2)^2 + O(x^30))) \\ _Colin Barker_, Jan 14 2020
%Y A331319 Cf. A322942 (Jacobsthal triangle), A331320, A331321.
%K A331319 nonn,easy
%O A331319 0,3
%A A331319 _Peter Luschny_, Jan 14 2020