A005557 Number of walks on square lattice.

42, 132, 297, 572, 1001, 1638, 2548, 3808, 5508, 7752, 10659, 14364, 19019, 24794, 31878, 40480, 50830, 63180, 77805, 95004, 115101, 138446, 165416, 196416, 231880, 272272, 318087, 369852, 428127, 493506, 566618, 648128, 738738, 839188, 950257, 1072764

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• easy

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 = 0..1000
• Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
• Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Sixth diagonal of Catalan triangle A033184.

Sixth column of Catalan triangle A009766.

Cf. A062991, A214292.

Programs

GAP

List([0..30],n->(n+1)*Binomial(n+10,4)/5); # Muniru A Asiru, Apr 10 2018

Magma

[(n+1)*Binomial(n+10, 4)/5: n in [0..40]]; // Vincenzo Librandi, Mar 20 2013

Maple

[seq(binomial(n,5)-binomial(n,3),n=9..55)]; # Zerinvary Lajos, Jul 19 2006
A005557:=(42-120*z+135*z**2-70*z**3+14*z**4)#(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

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 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{42,132,297,572,1001,1638},40] (* Harvey P. Dale, Feb 22 2024 *)

PARI

a(n)=(n+1)*binomial(n+10,4)/5 \\ Charles R Greathouse IV, Oct 21 2022

Formula

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

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.

a(n) = binomial(n+9,5) - binomial(n+9,3). - Zerinvary Lajos, Jul 19 2006

a(n) = A214292(n+9, 4). - Reinhard Zumkeller, Jul 12 2012

From Amiram Eldar, Sep 06 2022: (Start)

Sum_{n>=0} 1/a(n) = 2509/63504.

Sum_{n>=0} (-1)^n/a(n) = 951395/63504 - 1360*log(2)/63. (End)

Extensions

More terms and formula from Wolfdieter Lang, Sep 04 2001