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%I #35 Jun 14 2022 08:51:25
%S 1,3,6,20,75,336,1708,9792,62325,436480,3334386,27595776,245951615,
%T 2348666880,23923317720,258910994432,2966901358185,35886973648896,
%U 456927138333790,6108665873694720,85555744482868275,1252729007440396288,19140289332506060676
%N Third column of A162170.
%H J. A. Palacios, A. Bhaskar, F. Disanto and N. A. Rosenberg, <a href="https://doi.org/10.1007/s00285-022-01748-w">Enumeration of binary trees compatible with a perfect phylogeny</a>, J. Math. Biol. 84 (2022), 54.
%F a(n) = Sum_{k=1..floor(n/2)} (-1)^(floor(n/2)+k) * binomial(n+1, 2*k) * a(2*k-1) for n > 1. - _Mike Tryczak_, Jun 18 2015
%F a(n) = n*(n+1)/2 * A000111(n-1) (conjectured). - _Mike Tryczak_, Jun 17 2015
%F The above conjecture by Tryczak is correct. With an offset of 2, the e.g.f. is x^2/2!*(sec(x) + tan(x)). - _Peter Bala_, Sep 08 2021
%F a(n) is the number of ranked unlabeled binary tree shapes compatible with the binary perfect phylogeny (n,3). - _Noah A Rosenberg_, Jun 03 2022
%o (PARI) T(n, k) = if (k % 2, binomial(n-1, k-1) * (-1)^floor((n+k-1)/2), if (n==k, 1 , 0));
%o lista(nn) = {m = matrix(nn, nn, n, k, if (n>=k, T(n,k), 0)); m = m^(-1); for (n=3, nn, print1(m[n,3], ", "));} \\ _Michel Marcus_, Jun 17 2015
%o (PARI) lista(nn) = { a = [1]; for(n = 2, nn, a = concat(a, sum(k = 1, j = floor(n/2), (-1)^(j+k) * binomial(n+1, 2*k) * a[2*k-1]))); print(a) } \\ _Mike Tryczak_, Jun 18 2015
%Y Cf. A162170, A000111.
%K nonn,easy
%O 1,2
%A _Mats Granvik_, Jun 27 2009
%E Sequence corrected and extended by _Mike Tryczak_, Jun 17 2015
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