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A337863 Twice the total area of all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.

%I A337863 #18 Mar 24 2022 03:59:07
%S A337863 0,0,2,8,32,123,459,1715,6432,24256,91993,350817,1344559,5176106,
%T A337863 20003683,77567894,301667089,1176204195,4596267116,17995972170,
%U A337863 70581437130,277244854010,1090484851145,4294324521033,16929104620229,66801975748076,263827090563129
%N A337863 Twice the total area of all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.
%H A337863 Alois P. Heinz, <a href="/A337863/b337863.txt">Table of n, a(n) for n = 0..1663</a>
%p A337863 b:= proc(x, y) option remember; `if`(x=0, [1, 0],
%p A337863       add(add((p-> p+[0, p[1]*h*(2*y-v)])(b(x-h, y-v)),
%p A337863       h=1..min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))
%p A337863     end:
%p A337863 a:= n-> b(n, 0)[2]:
%p A337863 seq(a(n), n=0..30);
%t A337863 b[x_, y_] := b[x, y] = If[x == 0, {1, 0},
%t A337863      Sum[Sum[Function[p, p + {0, p[[1]]*h*(2*y - v)}][b[x - h, y - v]],
%t A337863      {h, 1, Min[x - y + v, Max[1, y - v]]}], {v, -1, Min[y, 1]}]];
%t A337863 a[n_] := b[n, 0][[2]];
%t A337863 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 24 2022, after _Alois P. Heinz_ *)
%Y A337863 Cf. A337067, A337318.
%K A337863 nonn
%O A337863 0,3
%A A337863 _Alois P. Heinz_, Oct 12 2020