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Royal paths in a lattice.
(Formerly M4200)
5

%I M4200 #36 Dec 24 2017 09:53:07

%S 1,6,30,146,714,3534,17718,89898,461010,2386390,12455118,65478978,

%T 346448538,1843520670,9859734630,52974158938,285791932578,

%U 1547585781414,8408765223294

%N Royal paths in a 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="/A006320/b006320.txt">Table of n, a(n) for n = 0..200</a>

%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).

%H G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

%H S.-n. Zheng and S.-l. Yang, <a href="http://dx.doi.org/10.1155/2014/848374">On the-Shifted Central Coefficients of Riordan Matrices</a>, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.

%F 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

%F a(n) = (3/n)*sum(binomial(n, j)*binomial(n+2+j, n-1), j=0..n) (n>0). - _Emeric Deutsch_, Aug 19 2004

%F 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

%F 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

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

%t 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 *)

%Y Third diagonal of A033877.

%Y Cf. A006318.

%K nonn

%O 0,2

%A _N. J. A. Sloane_