A006320 Royal paths in a lattice.

1, 6, 30, 146, 714, 3534, 17718, 89898, 461010, 2386390, 12455118, 65478978, 346448538, 1843520670, 9859734630, 52974158938, 285791932578, 1547585781414, 8408765223294, 45830521556466, 250501529133930, 1372777379874926, 7541129471504790, 41518462993275786

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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..200
• G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
• G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
• S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.

Crossrefs

Third diagonal of A033877.

Cf. A006318.

Programs

Maple

1,seq(3*sum(binomial(n,j)*binomial(n+2+j,n-1),j=0..n)/n,n=1..18);

Mathematica

Table[SeriesCoefficient[(1-x-Sqrt[1-6*x+x^2])^3/(8*x^3),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 05 2012 *)

Formula

3-fold convolution of the large Schroeder numbers (A006318). G.f.: R^3, where R = [1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of A006318. - Emeric Deutsch, Mar 15 2004

a(n) = (3/n)*sum(binomial(n, j)*binomial(n+2+j, n-1), j=0..n) (n>0). - Emeric Deutsch, Aug 19 2004

Recurrence: (n+3)*(5*n-1)*a(n) = 2*(15*n^2+20*n+13)*a(n-1) - (5*n^2+5*n-24)*a(n-2) + (n-3)*a(n-3). - Vaclav Kotesovec, Oct 05 2012

a(n) ~ 3 * (1 + sqrt(2))^(2*n+3) / (2^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 05 2012, simplified Dec 24 2017