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A383480 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(4,0),(0,1).

%I A383480 #19 May 29 2025 10:42:07
%S A383480 1,2,6,20,75,294,1176,4752,19350,79310,326898,1353768,5628441,
%T A383480 23478700,98217840,411879264,1730924700,7287941340,30736775190,
%U A383480 129825892000,549096132585,2325216522420,9857299586700,41830206233400,177673556967075,755307883986084,3213402383779812
%N A383480 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(4,0),(0,1).
%H A383480 Robert Israel, <a href="/A383480/b383480.txt">Table of n, a(n) for n = 0..1564</a>
%F A383480 a(n) = [x^n] 1/(1 - x - x^4)^(n+1).
%F A383480 a(n) = (n+1) * A063021(n+1).
%F A383480 a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,k) * binomial(2*n-3*k,n-4*k).
%p A383480 f:= proc(x,y) option remember;
%p A383480      local t;
%p A383480      t:= 0;
%p A383480      if x >= 1 then t:= t + procname(x-1,y) fi;
%p A383480      if x >= 4 then t:= t + procname(x-4,y) fi;
%p A383480      if y >= 1 then t:= t + procname(x,y-1) fi;
%p A383480      t
%p A383480 end proc:
%p A383480 f(0,0):= 1:
%p A383480 seq(f(n,n),n=0..26); # _Robert Israel_, May 28 2025
%o A383480 (PARI) a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(2*n-3*k, n-4*k));
%Y A383480 Cf. A038112, A383479.
%Y A383480 Cf. A063021, A383481.
%K A383480 nonn
%O A383480 0,2
%A A383480 _Seiichi Manyama_, Apr 28 2025