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

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

0,1

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).

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)

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 *)

Prog

(Magma) [(n+1)*Binomial(n+10, 4)/5: n in [0..40]]; // Vincenzo Librandi, Mar 20 2013
(GAP) List([0..30], n->(n+1)*Binomial(n+10, 4)/5); # Muniru A Asiru, Apr 10 2018
(PARI) a(n)=(n+1)*binomial(n+10, 4)/5 \\ Charles R Greathouse IV, Oct 21 2022

Crossrefs

Sixth diagonal of Catalan triangle A033184.

Sixth column of Catalan triangle A009766.

Cf. A062991, A214292.

Sequence in context: A298236 A299362 A304613 * A244102 A045088 A303860

Adjacent sequences: A005554 A005555 A005556 * A005558 A005559 A005560

Keyword

nonn,walk,easy

Author

N. J. A. Sloane

Extensions

More terms and formula from Wolfdieter Lang, Sep 04 2001

Status

approved