A264870 -id:A264870 - cp's OEIS frontend

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

Peter Bala, Nov 27 2015

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

			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)

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

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.

Cf. A086521, A264869, A264870, A291680.

Cf. A005773.

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);

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 *)

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

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

A086521 Number of tandem duplication trees on n duplicated gene segments.

Original entry on oeis.org

1, 1, 3, 11, 46, 210, 1021, 5202, 27477, 149324, 830357, 4705386, 27087106, 158019030, 932390694, 5555902302, 33391080001, 202196156448, 1232550473918, 7558030268270, 46592437224093, 288599067239678, 1795348952256896

Offset: 2

Michael D Hendy (m.hendy(AT)massey.ac.nz), Sep 10 2003

For n > 2, 2*a(n) is the number of rooted tandem duplication trees. See A264869.

			a(5) = 11, so there are 11 binary leaf labeled trees on 5 duplicate genes. As there are 15 binary leaf labeled trees, this means not all binary leaf labeled trees can represent a gene duplication tree.

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

O. Gascuel, M. Hendy, A. Jean-Marie and R. McLachlan, (2003) The combinatorics of tandem duplication trees, Systematic Biology 52, 110-118.
J. Yang and L. Zhang, On Counting Tandem Duplication Trees, Molecular Biology and Evolution, Volume 21, Issue 6, (2004) 1160-1163.

Cf. A264868, A264869, A264870.

with(combinat):
b := proc (n) option remember;
if n = 2 then 2 elif n = 3 then 2 else add((-1)^(k+1)*binomial(n+1-2*k, k)*b(n-k), k = 1..floor((n+1)/3)) end if; end proc:
seq(b(n)/2, n = 2..24); # Peter Bala, Nov 27 2015

a(n) = b(n+1, n-1), where b(n, 0) = b(n-1, 0) + b(n-1, 1); b(n, k) = b(n-1, k+1) + b(n, k-1), for k = 1, ..., n-2; with initial values b(2, 0) = 1, b(3, 0) = 0, b(3, 1) = 1.

For n >= 2, a(n) = b(n)/2, where b(n) = Sum_{k = 1..floor((n + 1)/3)} (-1)^(k + 1)*binomial(n + 1 - 2*k,k)*b(n-k) with b(1) = b(2) = 1 (Yang and Zhang). - Peter Bala, Nov 27 2015

More terms from David Wasserman, Mar 11 2005

cp's OEIS Frontend

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

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