A005562 Number of walks on square lattice. Column y=4 of A052174.

1, 5, 35, 140, 720, 2700, 12375, 45375, 196625, 715715, 3006003, 10930920, 45048640, 164105760, 668144880, 2441298600, 9859090500, 36149998500, 145173803500, 534239596880, 2136958387520, 7892175863000, 31479019635375, 116657543354625, 464342770607625, 1726402608669375

Offset: 4

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 = 4..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'(4).

Crossrefs

Cf. A005558, A005559, A005560, A005561, A093768.

Cf. A052174.

Programs

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

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:
A005562 := proc(n)
    wnprime(n,4) ;
end proc:
seq(A005562(n),n=4..30) ; # R. J. Mathar, Apr 02 2017

Mathematica

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 *)

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))}

Formula

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

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