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%I #48 Jan 29 2023 12:58:33
%S 0,0,0,1,2,6,20,69,246,894,3292,12242,45868,172884,654792,2489981,
%T 9500774,36356214,139471404,536217814,2065543012,7970227084,
%U 30801517624,119198827218,461863265660,1791626278060,6957151415832,27041349974436,105197526148312,409575623758440,1595836895778320
%N Number of binary trees with n internal nodes that contain the subtree [Z, [Z, U, U], [Z, U, U]].
%C The tree notation [Z, T1, T2] denotes an internal node and its two children T1 and T2. The notation U indicates a leaf.
%H Andrew Howroyd, <a href="/A331951/b331951.txt">Table of n, a(n) for n = 0..500</a>
%H Marko Riedel et al., Math Stackexchange, <a href="https://math.stackexchange.com/questions/3574610/">How many ways are there to count binary trees with superleaves</a>
%F G.f.: (sqrt(1 - 4*z + 4*z^4) - sqrt(1 - 4*z))/(2*z).
%F D-finite with recurrence -(n+1)*(n-3)*a(n) +(11*n^2-43*n+6)*a(n-1) +4*(-10*n^2+59*n-60)*a(n-2) +12*(2*n-5)*(2*n-11)*a(n-3) -4*(n-3)*(n-5)*a(n-4) +4*(7*n-29)*(n-6)*a(n-5) -24*(2*n-11)*(n-7)*a(n-6)=0. - _R. J. Mathar_, Mar 06 2022
%F a(n) = A000108(n+1)-A226022(n). - _R. J. Mathar_, Mar 06 2022
%e a(3) = 1 because from the five trees total, four paths do not contain the required subtree. These trees are:
%e . [Z, U, [Z, U, [Z, U, U]]],
%e . [Z, U, [Z, [Z, U, U], U]],
%e . [Z, [Z, U, U], [Z, U, U]], (*)
%e . [Z, [Z, U, [Z, U, U]], U],
%e . [Z, [Z, [Z, U, U], U], U].
%e The starred one is the one that contributes.
%o (PARI) seq(n)={Vec((sqrt(1 - 4*x + 4*x^4 + O(x*x^n)) - sqrt(1 - 4*x + O(x*x^n)))/(2*x), -n)} \\ _Andrew Howroyd_, Mar 11 2020
%Y Cf. A000108.
%K nonn,easy
%O 0,5
%A _Marko Riedel_, Mar 10 2020