A005560 Number of walks on square lattice. (Formerly M2987)

1, 3, 15, 45, 189, 588, 2352, 7560, 29700, 98010, 382239, 1288287, 5010005, 17177160, 66745536, 232092432, 901995588, 3173688180, 12342120700, 43861998180, 170724392916, 611947174608, 2384209771200, 8609646396000, 33577620944400, 122041737663300, 476432168185575

Offset

2,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

R. K. Guy, Letter to N. J. A. Sloane, May 1990

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6, w_n'(2).

Formula

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

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

Maple

wnprime := proc(n, y)
    local k;
    if type(n-y, 'even') then
        k := (n-y)/2 ;
        binomial(n+1, k)*(binomial(n, k)-binomial(n, k-1)) ;
    else
        k := (n-y-1)/2 ;
        binomial(n+1, k)*binomial(n, k+1)-binomial(n+1, k+1)*binomial(n, k-1) ;
    end if;
end proc:
A005560 := proc(n)
    wnprime(n, 2) ;
end proc:
seq(A005560(n), n=2..20) ; # R. J. Mathar, Apr 02 2017

Mathematica

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 *)

Prog

(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))}
(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

Crossrefs

Cf. A005558-A005562, A093768.

Sequence in context: A074355 A201868 A260021 * A100747 A100737 A178669

Adjacent sequences: A005557 A005558 A005559 * A005561 A005562 A005563

Keyword

nonn,walk

Author

N. J. A. Sloane

Status

approved