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%I #9 Jul 23 2024 10:49:22
%S 3,21,127,747,4386,25897,154077,923910,5581485,33949836,207787668,
%T 1278900412,7911394686,49165322241,306809507561,1921849861260,
%U 12079999018605,76170034283805,481680300300255,3054157623774495
%N Level of the first leaf (in preorder traversal) of a ternary tree, summed over all ternary trees with n edges. A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.
%C a(n) = Sum_{k=1..n} k*A121445(n,k).
%F a(n)=3n(23n^2+78n+67)binomial(3n+2,n+2)/[4(n+3)(2n+1)(2n+3)(2n+5)].
%F G.f.= (h-1-z)(h-1)/z^2, where h=1+zh^3=2sin(arcsin(sqrt(27z/4))/3)/sqrt(3z).
%F D-finite with recurrence -2*(2*n+5)*(n+3)*(1951*n-2094)*a(n) +(43553*n^3+142716*n^2+115045*n-10338)*a(n-1) +3*(2281*n+1723)*(3*n-1)*(3*n-2)*a(n-2)=0. - _R. J. Mathar_, Jul 24 2022
%e a(1)=3 because each of the trees /, | and \ contributes 1 to the sum.
%p a:=n->3*n*(23*n^2+78*n+67)*binomial(3*n+2,n+2)/4/(n+3)/(2*n+1)/(2*n+3)/(2*n+5): seq(a(n),n=1..23);
%Y Cf. A121445.
%K nonn
%O 1,1
%A _Emeric Deutsch_, Jul 30 2006