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A005560
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Number of walks on square lattice.
(Formerly M2987)
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5
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1, 3, 15, 45, 189, 588, 2352, 7560, 29700, 98010, 382239, 1288287, 5010005, 17177160, 66745536, 232092432, 901995588, 3173688180, 12342120700, 43861998180, 170724392916, 611947174608, 2384209771200, 8609646396000, 33577620944400, 122041737663300, 476432168185575
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OFFSET
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2,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 2..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'(2).
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FORMULA
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a(n) = C(n+3, ceiling(n/2))*C(n+2, floor(n/2)) - C(n+3, ceiling((n-1)/2))*C(n+2, floor((n-1)/2)). - Paul D. Hanna, Apr 16 2004
Conjecture: (n-1)*(n-2)*(2*n+1)*(n+5)*(n+4)*a(n) -4*n*(n+1)*(2*n^2+4*n+19)*a(n-1) -16*n^2*(n-1)*(2*n+3)*(n+1)*a(n-2)=0. - R. J. Mathar, Apr 02 2017
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MAPLE
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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:
A005560 := proc(n)
wnprime(n, 2) ;
end proc:
seq(A005560(n), n=2..20) ; # R. J. Mathar, Apr 02 2017
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MATHEMATICA
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Table[Binomial[n+3, Ceiling[n/2]] Binomial[n+2, Floor[n/2]]-Binomial[n+3, Ceiling[(n-1)/2]] Binomial[n+2, Floor[(n-1)/2]], {n, 0, 30}] (* Vincenzo Librandi, Apr 03 2017 *)
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PROG
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(PARI) {a(n)=binomial(n+3, ceil(n/2))*binomial(n+2, floor(n/2)) - binomial(n+3, ceil((n-1)/2))*binomial(n+2, floor((n-1)/2))}
(Magma) [Binomial(n+3, Ceiling(n/2))*Binomial(n+2, Floor(n/2)) - Binomial(n+3, Ceiling((n-1)/2))*Binomial(n+2, Floor((n-1)/2)): n in [0..30]]; // Vincenzo Librandi, Apr 03 2017
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CROSSREFS
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Cf. A005558-A005562, A093768.
Sequence in context: A074355 A201868 A260021 * A100747 A100737 A178669
Adjacent sequences: A005557 A005558 A005559 * A005561 A005562 A005563
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KEYWORD
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nonn,walk
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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