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A333678 Total area under all nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.

%I A333678 #15 Apr 26 2021 05:23:20
%S A333678 0,0,0,2,7,22,64,196,574,1762,5379,16378,49380,148892,449004,1353718,
%T A333678 4076150,12267160,36903433,110979048,333628384,1002722482,3013085711,
%U A333678 9052404522,27192329061,81671691634,245271884478,736513920180,2211445194899,6639545054310
%N A333678 Total area under all nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.
%H A333678 Alois P. Heinz, <a href="/A333678/b333678.txt">Table of n, a(n) for n = 0..250</a>
%H A333678 Alois P. Heinz, <a href="/A333647/a333647.gif">Animation of A333647(9) = 169 paths with total area a(9) = 1762</a>
%H A333678 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A333678 b:= proc(x, y, t) option remember; `if`(x=0, [1, 0],
%p A333678       add((p-> p+[0, p[1]*(y+j/2)])(b(x-1, y+j, j)),
%p A333678            j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
%p A333678     end:
%p A333678 a:= n-> b(n, 0$2)[2]:
%p A333678 seq(a(n), n=0..38);
%t A333678 b[x_, y_, t_] := b[x, y, t] = If[x == 0, {1, 0},
%t A333678      Sum[Function[p, p + {0, If[p === 0, 0, p[[1]]]*(y + j/2)}][
%t A333678      b[x-1, y+j, j]], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]];
%t A333678 a[n_] := b[n, 0, 0][[2]];
%t A333678 a /@ Range[0, 38] (* _Jean-François Alcover_, Apr 26 2021, after _Alois P. Heinz_ *)
%Y A333678 Cf. A333647, A333679, A333680.
%K A333678 nonn
%O A333678 0,4
%A A333678 _Alois P. Heinz_, Apr 01 2020