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A005560 Number of walks on square lattice. Column y=2 of A052174.

%I A005560 M2987 #39 Feb 03 2025 11:53:57
%S A005560 1,3,15,45,189,588,2352,7560,29700,98010,382239,1288287,5010005,
%T A005560 17177160,66745536,232092432,901995588,3173688180,12342120700,
%U A005560 43861998180,170724392916,611947174608,2384209771200,8609646396000,33577620944400,122041737663300,476432168185575
%N A005560 Number of walks on square lattice. Column y=2 of A052174.
%D A005560 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005560 Vincenzo Librandi, <a href="/A005560/b005560.txt">Table of n, a(n) for n = 2..1000</a>
%H A005560 R. K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>
%H A005560 R. K. Guy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6, w_n'(2).
%F A005560 a(n) = C(n+3, ceiling(n/2))*C(n+2, floor(n/2)) - C(n+3, ceiling((n-1)/2))*C(n+2, floor((n-1)/2)). - _Paul D. Hanna_, Apr 16 2004
%F A005560 Conjecture: (n-1)*(n-2)*(2*n+1)*(n+5)*(n+4)*a(n) -4*n*(n+1)*(2*n^2+4*n+19)*a(n-1) -16*n^2*(n-1)*(2*n+3)*(n+1)*a(n-2)=0. - _R. J. Mathar_, Apr 02 2017
%p A005560 wnprime := proc(n,y)
%p A005560     local k;
%p A005560     if type(n-y,'even') then
%p A005560         k := (n-y)/2 ;
%p A005560         binomial(n+1,k)*(binomial(n,k)-binomial(n,k-1)) ;
%p A005560     else
%p A005560         k := (n-y-1)/2 ;
%p A005560         binomial(n+1,k)*binomial(n,k+1)-binomial(n+1,k+1)*binomial(n,k-1) ;
%p A005560     end if;
%p A005560 end proc:
%p A005560 A005560 := proc(n)
%p A005560     wnprime(n,2) ;
%p A005560 end proc:
%p A005560 seq(A005560(n),n=2..20) ; # _R. J. Mathar_, Apr 02 2017
%t A005560 Table[Binomial[n+3, Ceiling[n/2]] Binomial[n+2, Floor[n/2]]-Binomial[n+3, Ceiling[(n-1)/2]] Binomial[n+2, Floor[(n-1)/2]], {n, 0, 30}] (* _Vincenzo Librandi_, Apr 03 2017 *)
%o A005560 (PARI) {a(n)=binomial(n+3,ceil(n/2))*binomial(n+2,floor(n/2)) - binomial(n+3,ceil((n-1)/2))*binomial(n+2,floor((n-1)/2))}
%o A005560 (Magma) [Binomial(n+3, Ceiling(n/2))*Binomial(n+2, Floor(n/2)) - Binomial(n+3, Ceiling((n-1)/2))*Binomial(n+2, Floor((n-1)/2)): n in [0..30]]; // _Vincenzo Librandi_, Apr 03 2017
%Y A005560 Cf. A005558, A005559, A005561, A005562, A093768.
%Y A005560 Cf. A052174.
%K A005560 nonn,walk
%O A005560 2,2
%A A005560 _N. J. A. Sloane_