A005561 Number of walks on square lattice. Column y=3 of A052174.

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

Author

N. J. A. Sloane

Keywords

• nonn
• walk

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).

Crossrefs

Cf. A005558, A005559, A005560, A005562, A093768.

Cf. A052174.

Programs

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

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 *)

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))}

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