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%I #8 Oct 14 2023 11:14:23
%S 1,7,55,319,1705,8238,37674,164229,692627,2843282,11433826,45212792,
%T 176385132,680452948,2600725892,9862321095,37150333241,139139984973,
%U 518538211261,1924077739700,7112221384554,26201080984497,96233327019085,352501632479306
%N Number of symmetrically unique Dyck paths of semilength n and height seven.
%H Alois P. Heinz, <a href="/A291891/b291891.txt">Table of n, a(n) for n = 7..1000</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (13, -60, 84, 203, -807, 569, 1173, -2090, 480, 1046, -662, -16, 76, -8).
%F G.f.: -x^7*(6*x-24*x^2+60*x^3-67*x^4+21*x^5+x^6+x^7-1) / ((x-1) *(2*x-1) *(2*x^2-4*x+1) *(x^3+3*x^2-1) *(x^3-9*x^2+6*x-1) *(2*x^4-4*x^2+1)).
%t CoefficientList[Series[-(6*x - 24*x^2 + 60*x^3 - 67*x^4 + 21*x^5 + x^6 + x^7 - 1)/((x - 1)*(2*x - 1)*(2*x^2 - 4*x + 1)*(x^3 + 3*x^2 - 1)*(x^3 - 9*x^2 + 6*x - 1)*(2*x^4 - 4*x^2 + 1)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Oct 14 2023 *)
%Y Column k=7 of A291883.
%K nonn,easy
%O 7,2
%A _Alois P. Heinz_, Sep 05 2017