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A247748 Total area below all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps u=(1,1), U=(1,3), H=(1,0), d=(1,-1) and D=(1,-3).

%I A247748 #14 Apr 29 2022 05:50:27
%S A247748 0,0,1,4,29,128,683,3048,15631,73584,367878,1766272,8772765,42778568,
%T A247748 212026492,1041894304,5164955809,25511089124,126587331646,
%U A247748 627213797064,3115744498691,15471098709336,76936461658184,382598697899276,1904401873987964,9480893249387192
%N A247748 Total area below all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps u=(1,1), U=(1,3), H=(1,0), d=(1,-1) and D=(1,-3).
%H A247748 Alois P. Heinz, <a href="/A247748/b247748.txt">Table of n, a(n) for n = 0..700</a>
%F A247748 a(n) = Sum_{k>=1} k * A247749(n,k).
%p A247748 b:= proc(x, y) option remember; `if`(y<0 or x<y, 0,
%p A247748       `if`(x=0, [1, 0], add((p->p+[0, p[1]*(y+j/2)])
%p A247748        (b(x-1, y+j)), j=[-1, -3, 0, 1, 3])))
%p A247748     end:
%p A247748 a:= n-> b(n, 0)[2]:
%p A247748 seq(a(n), n=0..30);
%t A247748 b[x_, y_] := b[x, y] = If[y < 0 || x < y, {0, 0},
%t A247748      If[x == 0, {1, 0}, Sum[Function[p, p + {0, p[[1]]*(y + j/2)}][
%t A247748      b[x - 1, y + j]], {j, {-1, -3, 0, 1, 3}}]]];
%t A247748 a[n_] := b[n, 0][[2]];
%t A247748 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 29 2022, after _Alois P. Heinz_ *)
%Y A247748 Cf. A240904, A247749.
%K A247748 nonn
%O A247748 0,4
%A A247748 _Alois P. Heinz_, Sep 23 2014