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A005561
Number of walks on square lattice. Column y=3 of A052174.
6
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
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
KEYWORD
nonn,walk
STATUS
approved