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

1, 4, 24, 84, 392, 1344, 5760, 19800, 81675, 283140, 1145144, 4008004, 16032016, 56632576, 225059328, 801773856, 3173688180, 11392726800, 44986664800, 162594659920, 641087516256, 2331227331840, 9183622822400, 33577620944400, 132211882468575, 485773975404900

Offset

3,2

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 = 3..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. See w_n'(3).

Formula

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

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

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:
A005561 := proc(n)
    wnprime(n, 3) ;
end proc:
seq(A005561(n), n=3..30) ; # R. J. Mathar, Apr 02 2017

Mathematica

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 *)

Prog

(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))}
(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

Crossrefs

Cf. A005558-A005562, A093768.

Sequence in context: A334581 A341688 A341877 * A061612 A097875 A212681

Adjacent sequences: A005558 A005559 A005560 * A005562 A005563 A005564

Keyword

nonn,walk

Author

N. J. A. Sloane

Status

approved