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A330169 a(n) is the total area of all closed Deutsch paths of length n.

%I A330169 #59 Dec 19 2024 06:17:24
%S A330169 1,3,12,39,129,411,1300,4065,12633,39046,120204,368844,1128837,
%T A330169 3447303,10508592,31985085,97226733,295214316,895502520,2714106318,
%U A330169 8219809425,24877611798,75248738292,227488953354,687408882709,2076269682831,6268788729240,18920387069731,57086882549253
%N A330169 a(n) is the total area of all closed Deutsch paths of length n.
%C A330169 Deutsch paths are a variation of Dyck paths that allow for down-steps of arbitrary length.
%H A330169 Alois P. Heinz, <a href="/A330169/b330169.txt">Table of n, a(n) for n = 2..2096</a>
%H A330169 Helmut Prodinger, <a href="https://arxiv.org/abs/2003.01918">Deutsch paths and their enumeration</a>, arXiv:2003.01918 [math.CO], 2020. See p. 8.
%F A330169 G.f.: v^2*(1+v+v^2)^2/((1+v)^3*(1-v)^2) where v=(1-z-sqrt(1-2*z-3*z^2))/(2*z), that is, where v is the g.f. of A001006.
%p A330169 a:= proc(n) option remember;`if`(n<4, [0$2, 1, 3][n+1], (4*n*
%p A330169       a(n-1)+(2*n+4)*a(n-2)+12*(1-n)*a(n-3)+9*(1-n)*a(n-4))/(n+1))
%p A330169     end:
%p A330169 seq(a(n), n=2..30);  # _Alois P. Heinz_, Mar 05 2020
%t A330169 a = DifferenceRoot[Function[{y, n}, {9(n+3)y[n] + 12(n+3)y[n+1] - 2(n+6)y[n+2] - 4(n+4)y[n+3] + (n+5)y[n+4] == 0, y[2] == 1, y[3] == 3, y[4] == 12, y[5] == 39}]];
%t A330169 a /@ Range[2, 30] (* _Jean-François Alcover_, Mar 12 2020 *)
%o A330169 (PARI) my(z='z+O('z^30), v=(1-z-sqrt(1-2*z-3*z^2))/(2*z)); Vec(v^2*(1+v+v^2)^2/((1+v)^3*(1-v)^2))
%Y A330169 Cf. A001006 (Motzkin numbers), A005043, A333017, A333098.
%K A330169 nonn
%O A330169 2,2
%A A330169 _Michel Marcus_, Mar 05 2020