|
%I #28 Jul 01 2020 15:23:09
%S 1,3,9,36,162,783,3969,20817,112023,615033,3431403,19398690,110880900,
%T 639730305,3720657807,21790419444,128398625658,760668489729,
%U 4528069760691,27070491820644,162464919528222,978463778897637
%N Expansion of (1-x-sqrt(1-6x-3x^2))/(2x).
%C Transform of Catalan numbers by Riordan array ((1+x)/(1-x), x(1+x)/(1-x)^2).
%H Vincenzo Librandi, <a href="/A156016/b156016.txt">Table of n, a(n) for n = 0..200</a>
%H M. Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
%H M. Dziemianczuk, <a href="https://doi.org/10.1016/j.disc.2015.11.001">On Directed Lattice Paths With Additional Vertical Steps</a>, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
%F a(n) = Sum_{k=0..n} Sum_{j=0..k+1} C(k+1,j)*C(n+k-j,n-k-j)*A000108(k).
%F a(n+1) = 3*A107264(n-1). - _Philippe Deléham_, Feb 04 2009
%F D-finite with recurrence: (n+1)*a(n) + 3*(-2*n+1)*a(n-1) + 3*(-n+2)*a(n-2) = 0. - _R. J. Mathar_, Dec 03 2014
%F G.f. A(x) satisfies: A(x) = 1 + x * (1 + A(x) + A(x)^2). - _Ilya Gutkovskiy_, Jul 01 2020
%t CoefficientList[Series[(1-x-Sqrt[1-6x-3x^2])/(2x),{x,0,30}],x] (* _Harvey P. Dale_, Jul 27 2014 *)
%Y Cf. A000108, A100239, A107264.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Feb 01 2009
|
