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A005561
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Number of walks on square lattice.
(Formerly M3596)
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5
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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
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OFFSET
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3,2
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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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FORMULA
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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
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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:
wnprime(n, 3) ;
end proc:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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