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A331106 Number of plane trees of total weight n of combinatorial class T=Z*U + Z*T^2/(1-T) with nodes Z of weight one and leaves U of weight three.

%I A331106 #27 Dec 19 2024 21:18:22
%S A331106 0,0,0,1,0,0,0,0,1,0,0,0,1,2,0,0,1,5,5,0,1,9,21,14,1,14,56,84,43,20,
%T A331106 120,300,331,159,225,825,1486,1322,814,1925,5006,7051,5621,5434,14015,
%U A331106 28082,32968,27092,39261,91793,149877,156858,152023,276769,558845,778920,786931,953756
%N A331106 Number of plane trees of total weight n of combinatorial class T=Z*U + Z*T^2/(1-T) with nodes Z of weight one and leaves U of weight three.
%C A331106 The underlying tree structure before the weights are applied (Z with weight one, U with weight three) is a series-reduced tree because a non-leaf node Z has at least two children.
%D A331106 P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009.
%H A331106 Marko Riedel et al., Math.StackExchange, <a href="https://math.stackexchange.com/questions/3499909/">Counting plane trees</a>
%F A331106 G.f.: (1 + z^4 - sqrt(z^8 - 4*z^5 - 2*z^4 +1))/(2*(z+1)).
%F A331106 a(n) = Sum_{k=floor(n/5)+1..floor((n-1)/4)} (1/(n-3*k)) * binomial(n-3*k,k) * binomial(k-2, n-4*k-1) for n >= 1, n <> 4. a(4) = 1.
%F A331106 D-finite with recurrence: n*a(n) +(n)*a(n-1) +(n-2)*a(n-2) +(n-2)*a(n-3) +2*(-n+6)*a(n-4) +6*(-n+7)*a(n-5) +2*(-3*n+23)*a(n-6) +6*(-n+9)*a(n-7) +(-3*n+26)*a(n-8) +(n-12)*a(n-9) +(n-14)*a(n-10) +(n-14)*a(n-11)=0. - _R. J. Mathar_, Jan 27 2020
%e A331106 For n=4, the tree is Z-U, for n=9 the tree is
%e A331106     Z-U
%e A331106    /
%e A331106   Z
%e A331106    \
%e A331106     Z-U.
%Y A331106 Cf. A308616.
%K A331106 nonn
%O A331106 1,14
%A A331106 _Marko Riedel_, Jan 09 2020