%I M5277 #71 Feb 22 2024 19:15:28
%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.
%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 Richard K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>.
%H Richard 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="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%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 a(n) = binomial(n+9,5) - binomial(n+9,3). - _Zerinvary Lajos_, Jul 19 2006
%F a(n) = A214292(n+9, 4). - _Reinhard Zumkeller_, Jul 12 2012
%F From _Amiram Eldar_, Sep 06 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 2509/63504.
%F Sum_{n>=0} (-1)^n/a(n) = 951395/63504 - 1360*log(2)/63. (End)
%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 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{42,132,297,572,1001,1638},40] (* _Harvey P. Dale_, Feb 22 2024 *)
%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
%o (PARI) a(n)=(n+1)*binomial(n+10,4)/5 \\ _Charles R Greathouse IV_, Oct 21 2022
%Y Sixth diagonal of Catalan triangle A033184.
%Y Sixth column of Catalan triangle A009766.
%Y Cf. A062991, A214292.
%K nonn,walk,easy
%O 0,1
%A _N. J. A. Sloane_
%E More terms and formula from _Wolfdieter Lang_, Sep 04 2001