A162171 Third column of A162170.

1, 3, 6, 20, 75, 336, 1708, 9792, 62325, 436480, 3334386, 27595776, 245951615, 2348666880, 23923317720, 258910994432, 2966901358185, 35886973648896, 456927138333790, 6108665873694720, 85555744482868275, 1252729007440396288, 19140289332506060676

Offset

1,2

Links

Table of n, a(n) for n=1..23.

J. A. Palacios, A. Bhaskar, F. Disanto and N. A. Rosenberg, Enumeration of binary trees compatible with a perfect phylogeny, J. Math. Biol. 84 (2022), 54.

Formula

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

a(n) = n*(n+1)/2 * A000111(n-1) (conjectured). - Mike Tryczak, Jun 17 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

Prog

(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

Crossrefs

Cf. A162170, A000111.

Sequence in context: A168594 A123559 A334329 * A194992 A163881 A238474

Adjacent sequences: A162168 A162169 A162170 * A162172 A162173 A162174

Keyword

nonn,easy

Author

Mats Granvik, Jun 27 2009

Extensions

Sequence corrected and extended by Mike Tryczak, Jun 17 2015

Status

approved