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A108430 Number of d steps in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).

%I A108430 #13 Jul 24 2022 11:18:58
%S A108430 3,31,311,3151,32299,334335,3488239,36627487,386618387,4098713631,
%T A108430 43611791783,465496885231,4981942135611,53443871159551,
%U A108430 574500093677535,6186886528903231,66735614131858723,720897596248427295
%N A108430 Number of d steps in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).
%H A108430 Emeric Deutsch, <a href="http://www.jstor.org/stable/2589192">Problem 10658: Another Type of Lattice Path</a>, American Math. Monthly, 107, 2000, 368-370.
%F A108430 a(n) = (1/n)*sum(k*binomial(n,2n-k)*binomial(n+k,n-1), k=n..2n).
%F A108430 Conjecture D-finite with recurrence n*(2*n+1)*(23982*n-28681)*a(n) +(-640736*n^3+1168048*n^2-901220*n+247035)*a(n-1) +(1196488*n^3-6448608*n^2+10992587*n-5911365)*a(n-2) +2*(2*n-5)*(28283*n-6993)*(n-3)*a(n-3)=0. - _R. J. Mathar_, Jul 24 2022
%e A108430 a(1) = 3 because in the paths ud, Udd we have 3 d steps altogether.
%p A108430 a:=n->(1/n)*sum(k*binomial(n,2*n-k)*binomial(n+k,n-1),k=n..2*n): seq(a(n),n=1..22);
%Y A108430 Cf. A027307, A108429.
%K A108430 nonn
%O A108430 1,1
%A A108430 _Emeric Deutsch_, Jun 03 2005