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A331951 Number of binary trees with n internal nodes that contain the subtree [Z, [Z, U, U], [Z, U, U]]. 0
0, 0, 0, 1, 2, 6, 20, 69, 246, 894, 3292, 12242, 45868, 172884, 654792, 2489981, 9500774, 36356214, 139471404, 536217814, 2065543012, 7970227084, 30801517624, 119198827218, 461863265660, 1791626278060, 6957151415832, 27041349974436, 105197526148312, 409575623758440, 1595836895778320 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The tree notation [Z, T1, T2] denotes an internal node and its two children T1 and T2. The notation U indicates a leaf.

LINKS

Table of n, a(n) for n=0..30.

Marko Riedel et al., Math Stackexchange, How many ways are there to count binary trees with superleaves

FORMULA

G.f.: (sqrt(1 - 4*z + 4*z^4) - sqrt(1 - 4*z))/(2*z).

EXAMPLE

a(3) = 1 because from the five trees total, four paths do not contain the required subtree. These trees are:

.    [Z, U, [Z, U, [Z, U, U]]],

.    [Z, U, [Z, [Z, U, U], U]],

.    [Z, [Z, U, U], [Z, U, U]], (*)

.    [Z, [Z, U, [Z, U, U]], U],

.    [Z, [Z, [Z, U, U], U], U].

The starred one is the one that contributes.

PROG

(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

CROSSREFS

Cf. A000108.

Sequence in context: A026029 A078483 A163135 * A047036 A199248 A148478

Adjacent sequences:  A331948 A331949 A331950 * A331952 A331953 A331954

KEYWORD

nonn

AUTHOR

Marko Riedel, Mar 10 2020

STATUS

approved

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Last modified December 5 03:59 EST 2021. Contains 349530 sequences. (Running on oeis4.)