%I M3974 #41 Sep 08 2022 08:44:33
%S 1,5,35,140,720,2700,12375,45375,196625,715715,3006003,10930920,
%T 45048640,164105760,668144880,2441298600,9859090500,36149998500,
%U 145173803500,534239596880,2136958387520,7892175863000,31479019635375,116657543354625,464342770607625,1726402608669375
%N Number of walks on square lattice.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005562/b005562.txt">Table of n, a(n) for n = 4..1000</a>
%H R. K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>
%H R. K. Guy, <a href="http://www.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'(4)
%F a(n) = C(n+5, ceiling(n/2))*C(n+4, floor(n/2)) - C(n+5, ceiling((n-1)/2))*C(n+4, floor((n-1)/2)). - _Paul D. Hanna_, Apr 16 2004
%F Conjecture: (n-3)*(n-4)*(2*n+1)*(n+7)*(n+6)*a(n) - 4*n*(n+1)*(2*n^2+4*n+51)*a(n-1) - 16*n^2*(n-1)*(2*n+3)*(n+1)*a(n-2) = 0. - _R. J. Mathar_, Apr 02 2017
%p wnprime := proc(n,y)
%p local k;
%p if type(n-y,'even') then
%p k := (n-y)/2 ;
%p binomial(n+1,k)*(binomial(n,k)-binomial(n,k-1)) ;
%p else
%p k := (n-y-1)/2 ;
%p binomial(n+1,k)*binomial(n,k+1)-binomial(n+1,k+1)*binomial(n,k-1) ;
%p end if;
%p end proc:
%p A005562 := proc(n)
%p wnprime(n,4) ;
%p end proc:
%p seq(A005562(n),n=4..30) ; # _R. J. Mathar_, Apr 02 2017
%t Table[Binomial[n+5, Ceiling[n/2]] Binomial[n+4, Floor[n/2]]-Binomial[n+5, Ceiling[(n-1)/2]] Binomial[n+4, Floor[(n-1)/2]], {n, 0, 30}] (* _Vincenzo Librandi_, Apr 03 2017 *)
%o (PARI) {a(n)=binomial(n+5,ceil(n/2))*binomial(n+4,floor(n/2)) - binomial(n+5,ceil((n-1)/2))*binomial(n+4,floor((n-1)/2))}
%o (Magma) [Binomial(n+5, Ceiling(n/2))*Binomial(n+4, Floor(n/2)) - Binomial(n+5, Ceiling((n-1)/2))*Binomial(n+4, Floor((n-1)/2)): n in [0..30]]; // _Vincenzo Librandi_, Apr 03 2017
%Y Cf. A005558, A005559, A005560, A005561, A093768.
%K nonn,walk
%O 4,2
%A _N. J. A. Sloane_