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A005561 Number of walks on square lattice.
(Formerly M3596)

%I M3596

%S 1,4,24,84,392,1344,5760,19800,81675,283140,1145144,4008004,16032016,

%T 56632576,225059328,801773856,3173688180,11392726800,44986664800,

%U 162594659920,641087516256,2331227331840,9183622822400,33577620944400,132211882468575,485773975404900

%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="/A005561/b005561.txt">Table of n, a(n) for n = 3..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="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. See w_n'(3).

%F a(n) = C(n+4, ceiling(n/2))*C(n+3, floor(n/2)) - C(n+4, ceiling((n-1)/2))*C(n+3, floor((n-1)/2)). - _Paul D. Hanna_, Apr 16 2004

%F Conjecture: (n-2)*(n-3)*(2*n+1)*(n+6)*(n+5)*a(n) - 4*n*(n+1)*(2*n^2+4*n+33)*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 A005561 := proc(n)

%p wnprime(n,3) ;

%p end proc:

%p seq(A005561(n),n=3..30) ; # _R. J. Mathar_, Apr 02 2017

%t Table[Binomial[n+4, Ceiling[n/2]] Binomial[n+3, Floor[n/2]]-Binomial[n+4, Ceiling[(n-1)/2]] Binomial[n+3, Floor[(n-1)/2]], {n, 0, 30}] (* _Vincenzo Librandi_, Apr 03 2017 *)

%o (PARI) {a(n)=binomial(n+4,ceil(n/2))*binomial(n+3,floor(n/2)) - binomial(n+4,ceil((n-1)/2))*binomial(n+3,floor((n-1)/2))}

%o (MAGMA) [Binomial(n+4, Ceiling(n/2))*Binomial(n+3, Floor(n/2)) - Binomial(n+4, Ceiling((n-1)/2))*Binomial(n+3, Floor((n-1)/2)): n in [0..30]]; // _Vincenzo Librandi_, Apr 03 2017

%Y Cf. A005558-A005562, A093768.

%K nonn,walk

%O 3,2

%A _N. J. A. Sloane_

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Last modified December 5 21:00 EST 2019. Contains 329779 sequences. (Running on oeis4.)