A005562 Number of walks on square lattice. (Formerly M3974)

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

4,2

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 = 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)

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

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 *)

Prog

(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))}
(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

Crossrefs

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

Sequence in context: A096743 A026697 A000910 * A097872 A184707 A124793

Adjacent sequences: A005559 A005560 A005561 * A005563 A005564 A005565

Keyword

nonn,walk

Author

N. J. A. Sloane

Status

approved