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A274373 Sum of the areas of all modified skew Dyck paths of semilength n.

%I A274373 #13 Dec 29 2020 09:03:55
%S A274373 0,1,6,35,188,989,5131,26411,135229,689814,3509014,17811637,90256685,
%T A274373 456719880,2308440442,11656409995,58809893357,296500180806,
%U A274373 1493924791698,7523064390774,37866103978109,190510720248534,958122016323881,4816944544836927,24209532464417585
%N A274373 Sum of the areas of all modified skew Dyck paths of semilength n.
%C A274373 A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.
%C A274373 a(n)^(1/n) tends to 5. - _Vaclav Kotesovec_, Jun 26 2016
%H A274373 Alois P. Heinz, <a href="/A274373/b274373.txt">Table of n, a(n) for n = 0..500</a>
%F A274373 a(n) = Sum_{k=n..n^2} k * A274372(n,k).
%e A274373 a(3) = 35 = 9+7+5+6+5+3 = sum of the areas of UUUDDD, UUDUDD, UUDDUD, UAUDDD, UDUUDD, UDUDUD, respectively.
%p A274373 b:= proc(x, y, t, n) option remember; `if`(y>n, 0, `if`(n=y,
%p A274373      `if`(t=2, 0, [1, 0]), (p-> p+[0, p[1]*(2*y+1)])(b(x+1, y
%p A274373       +1, 0, n-1))+`if`(t<>1 and x>0, b(x-1, y+1, 2, n-1), 0)
%p A274373       +`if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0)))
%p A274373     end:
%p A274373 a:= n-> b(0$3, 2*n)[2]:
%p A274373 seq(a(n), n=0..30);
%t A274373 b[x_, y_, t_, n_] := b[x, y, t, n] = If[y > n, 0, If[n == y, If[t == 2, {0, 0}, {1, 0}], Function[p, p + {0, p[[1]] (2y + 1)}][b[x + 1, y + 1, 0, n - 1]] + If[t != 1 && x > 0, b[x - 1, y + 1, 2, n - 1], 0] + If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]];
%t A274373 a[n_] := b[0, 0, 0, 2 n][[2]];
%t A274373 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A274373 Cf. A230823, A274372.
%K A274373 nonn
%O A274373 0,3
%A A274373 _Alois P. Heinz_, Jun 19 2016