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A333682 Number of nonnegative lattice paths from (0,0) to (4n+3,0) such that slopes of adjacent steps differ by one, assuming zero slope before and after the paths.

%I A333682 #17 Apr 03 2020 18:11:23
%S A333682 1,3,16,119,1070,10751,116287,1326581,15756587,193181910,2429921124,
%T A333682 31216684816,408198225495,5418728779290,72871393962150,
%U A333682 991102308239835,13613940451015378,188650695857473559,2634681336798911129,37054660535787380825,524449965598846642847
%N A333682 Number of nonnegative lattice paths from (0,0) to (4n+3,0) such that slopes of adjacent steps differ by one, assuming zero slope before and after the paths.
%C A333682 The maximal height in all paths of length 4n+3 is (n+1)^2 = A000290(n+1).
%C A333682 The maximal area under all paths of length 4n+3 is 2*(n+1)^3 = A033431(n+1).
%H A333682 Alois P. Heinz, <a href="/A333682/b333682.txt">Table of n, a(n) for n = 0..100</a>
%H A333682 Alois P. Heinz, <a href="/A333682/a333682.gif">Animation of a(3) = 119 paths</a>
%H A333682 Alois P. Heinz, <a href="/A333682/a333682_1.gif">Plot of a(3) = 119 paths</a>
%H A333682 Alois P. Heinz, <a href="/A333682/a333682_2.gif">Plot of a(4) = 1070 paths</a>
%H A333682 Alois P. Heinz, <a href="/A333682/a333682_3.gif">Plot of a(5) = 10751 paths</a>
%p A333682 b:= proc(x, y, t) option remember; `if`(x=0, 1, add(`if`(j=t, 0,
%p A333682       b(x-1, y+j, j)), j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
%p A333682     end:
%p A333682 a:= n-> b(4*n+3, 0$2):
%p A333682 seq(a(n), n=0..23);
%Y A333682 Cf. A000290, A004767, A033431, A333647.
%K A333682 nonn
%O A333682 0,2
%A A333682 _Alois P. Heinz_, Apr 01 2020