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A383478 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(2,0),(3,0),(0,1).

%I A383478 #16 May 28 2025 16:22:41
%S A383478 1,2,9,44,220,1134,5950,31608,169488,915420,4972825,27141036,
%T A383478 148711836,817516196,4506838380,24906188912,137933068572,765324011736,
%U A383478 4253501563156,23675286219200,131955035141160,736347621539310,4113559552360230,23003228298637080
%N A383478 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(2,0),(3,0),(0,1).
%H A383478 Robert Israel, <a href="/A383478/b383478.txt">Table of n, a(n) for n = 0..1316</a>
%F A383478 a(n) = [x^n] 1/(1 - x - x^2 - x^3)^(n+1).
%F A383478 a(n) = (n+1) * A063018(n+1).
%p A383478 f:= proc(x,y) option remember;
%p A383478      local t;
%p A383478      t:= 0;
%p A383478      if x >= 1 then t:= t + procname(x-1,y) fi;
%p A383478      if x >= 2 then t:= t + procname(x-2,y) fi;
%p A383478      if x >= 3 then t:= t + procname(x-3,y) fi;
%p A383478      if y >= 1 then t:= t + procname(x,y-1) fi;
%p A383478      t
%p A383478 end proc:
%p A383478 f(0,0):= 1:
%p A383478 seq(f(n,n),n=0..25); # _Robert Israel_, May 28 2025
%Y A383478 Main diagonal of A383477.
%Y A383478 Cf. A063018.
%K A383478 nonn
%O A383478 0,2
%A A383478 _Seiichi Manyama_, Apr 28 2025