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A005557 Number of walks on square lattice.
(Formerly M5277)
7

%I M5277

%S 42,132,297,572,1001,1638,2548,3808,5508,7752,10659,14364,19019,24794,

%T 31878,40480,50830,63180,77805,95004,115101,138446,165416,196416,

%U 231880,272272,318087,369852,428127,493506,566618,648128,738738,839188,950257,1072764

%N Number of walks on square lattice.

%C a(n) = A214292(n+9, 4). - _Reinhard Zumkeller_, Jul 12 2012

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A005557/b005557.txt">Table of n, a(n) for n = 0..1000</a>

%H R. K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>

%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%F a(n) = A009766(n+5, 5) = (n+1)*binomial(n+10, 4)/5.

%F G.f.: (42 - 120*x + 135*x^2 - 70*x^3 + 14*x^4)/(1-x)^6; numerator polynomial is N(2;4, x) from A062991.

%F Binomial(n,5) - binomial(n,3), n >= 9. - _Zerinvary Lajos_, Jul 19 2006

%p [seq(binomial(n,5)-binomial(n,3),n=9..55)]; # _Zerinvary Lajos_, Jul 19 2006

%p A005557:=(42-120*z+135*z**2-70*z**3+14*z**4)#(z-1)**6; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t CoefficientList[Series[(14 z^4 - 70 z^3 + 135 z^2 - 120 z + 42)/(z - 1)^6, {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 22 2011 *)

%o (MAGMA) [(n+1)*Binomial(n+10, 4)/5: n in [0..40]]; // _Vincenzo Librandi_, Mar 20 2013

%o (GAP) List([0..30],n->(n+1)*Binomial(n+10,4)/5); # _Muniru A Asiru_, Apr 10 2018

%Y Sixth diagonal of Catalan triangle A033184. Sixth column of Catalan triangle A009766.

%K nonn,walk

%O 0,1

%A _N. J. A. Sloane_

%E More terms and formula from _Wolfdieter Lang_, Sep 04 2001

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Last modified February 20 16:42 EST 2020. Contains 332080 sequences. (Running on oeis4.)