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1, 3, 6, 20, 75, 336, 1708, 9792, 62325, 436480, 3334386, 27595776, 245951615, 2348666880, 23923317720, 258910994432, 2966901358185, 35886973648896, 456927138333790, 6108665873694720, 85555744482868275, 1252729007440396288, 19140289332506060676
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refs;
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internal format)
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OFFSET
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1,2
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LINKS
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FORMULA
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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
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
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
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PROG
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(PARI) T(n, k) = if (k % 2, binomial(n-1, k-1) * (-1)^floor((n+k-1)/2), if (n==k, 1 , 0));
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
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Sequence corrected and extended by Mike Tryczak, Jun 17 2015
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STATUS
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approved
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