A264868 Number of rooted tandem duplication trees on n gene segments.

1, 1, 2, 6, 22, 92, 420, 2042, 10404, 54954, 298648, 1660714, 9410772, 54174212, 316038060, 1864781388, 11111804604, 66782160002, 404392312896, 2465100947836, 15116060536540, 93184874448186, 577198134479356, 3590697904513792, 22425154536754776

Offset: 1

Author

Peter Bala, Nov 27 2015

Keywords

• nonn
• easy

Comments

Apparently a(n) is the number of words [d(0)d(1)d(2)...d(n)] where d(k) <= k (so d(0)=0) and if w(k-1) > w(k) then w(k-1) - w(k) = 1 (that is, descents by 2 or more are forbidden). - Joerg Arndt, Jan 26 2024

Examples

			Form _Joerg Arndt_, Jan 26 2024: (Start)
The a(5) = 22 words as described in the comment are (dots denote zeros, leading zeros omitted):
    1:  [ . . . ]
    2:  [ . . 1 ]
    3:  [ . . 2 ]
    4:  [ . . 3 ]
    5:  [ . 1 . ]
    6:  [ . 1 1 ]
    7:  [ . 1 2 ]
    8:  [ . 1 3 ]
    9:  [ . 2 1 ]
   10:  [ . 2 2 ]
   11:  [ . 2 3 ]
   12:  [ 1 . . ]
   13:  [ 1 . 1 ]
   14:  [ 1 . 2 ]
   15:  [ 1 . 3 ]
   16:  [ 1 1 . ]
   17:  [ 1 1 1 ]
   18:  [ 1 1 2 ]
   19:  [ 1 1 3 ]
   20:  [ 1 2 1 ]
   21:  [ 1 2 2 ]
   22:  [ 1 2 3 ]
(End)
		

References

• Mathematics of Evolution and Phylogeny, O. Gascuel (ed.), Oxford University Press, 2005

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• O. Gascuel, M. Hendy, A. Jean-Marie and R. McLachlan, The combinatorics of tandem duplication trees, Systematic Biology 52, (2003), 110-118.
• J. Yang and L. Zhang, Letter. On Counting Tandem Duplication Trees, Molecular Biology and Evolution, Volume 21, Issue 6, (2004) 1160-1163.

Crossrefs

Cf. A086521, A264869, A264870, A291680.

Cf. A005773.

Programs

Maple

a:= proc(n) option remember;
       if n = 1 then 1 elif n = 2 then 1 else add((-1)^(k+1)*
          binomial(n+1-2*k, k)*a(n-k), k = 1..floor((n+1)/3))
       end if;
    end proc:
seq(a(n), n = 1..24);

Mathematica

a[n_] := a[n] = If[n == 1, 1, If[n == 2, 1, Sum[(-1)^(k+1) Binomial[n+1-2k, k] a[n-k], {k, 1, Floor[(n+1)/3]}]]]; Array[a, 25] (* Jean-François Alcover, May 29 2019 *)

Python

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def a(n):
    return 1 if n<3 else sum([(-1)**(k + 1)*binomial(n + 1 - 2*k, k)*a(n - k) for k in range(1, (n + 1)//3 + 1)])
print([a(n) for n in range(1, 26)]) # Indranil Ghosh, Aug 30 2017

Formula

a(n) = Sum_{k = 1..floor((n + 1)/3)} (-1)^(k + 1)*binomial(n + 1 - 2*k,k)*a(n-k) with a(1) = a(2) = 1 (Yang and Zhang).

For n >= 3, (1/2)*a(n) = A086521(n) is the number of tandem duplication trees on n gene segments.

Main diagonal and row sums of A264869.

a(n) = Sum_{k=0..n-1} A291680(n-1,k). - Alois P. Heinz, Aug 29 2017