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A006320 Royal paths in a lattice.

%I A006320 M4200 #40 Nov 26 2024 08:32:30
%S A006320 1,6,30,146,714,3534,17718,89898,461010,2386390,12455118,65478978,
%T A006320 346448538,1843520670,9859734630,52974158938,285791932578,
%U A006320 1547585781414,8408765223294,45830521556466,250501529133930,1372777379874926,7541129471504790,41518462993275786
%N A006320 Royal paths in a lattice.
%D A006320 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006320 Vincenzo Librandi, <a href="/A006320/b006320.txt">Table of n, a(n) for n = 0..200</a>
%H A006320 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 A006320 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 A006320 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 A006320 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 A006320 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 A006320 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 A006320 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 A006320 1,seq(3*sum(binomial(n,j)*binomial(n+2+j,n-1),j=0..n)/n,n=1..18);
%t A006320 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 A006320 Third diagonal of A033877.
%Y A006320 Cf. A006318.
%K A006320 nonn
%O A006320 0,2
%A A006320 _N. J. A. Sloane_