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 A005560 Number of walks on square lattice. (Formerly M2987) 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 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 = 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). FORMULA 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 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: A005560 := proc(n) wnprime(n, 2) ; end proc: seq(A005560(n), n=2..20) ; # R. J. Mathar, Apr 02 2017 MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A005558-A005562, A093768. Sequence in context: A074355 A201868 A260021 * A100747 A100737 A178669 Adjacent sequences: A005557 A005558 A005559 * A005561 A005562 A005563 KEYWORD nonn,walk AUTHOR STATUS approved

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Last modified March 29 06:37 EDT 2023. Contains 361596 sequences. (Running on oeis4.)