A380766 - OEIS

%I #15 Feb 09 2025 22:58:21

%S 1,3,3,9,11,9,28,37,37,28,90,124,134,124,90,297,420,473,473,420,297,

%T 1001,1441,1665,1735,1665,1441,1001,3432,5005,5885,6291,6291,5885,

%U 5005,3432,11934,17576,20930,22766,23354,22766,20930,17576,11934,41990,62322,74932,82537,86149,86149,82537,74932,62322,41990

%N Triangle read by rows: T(n,k) is the number of coalescent histories for an n-leaf caterpillar species tree and an identically labeled k-pseudocaterpillar gene tree, 3 <= k <= n.

%C The largest entry in a row with fixed n is at k=(n+3)/2 for n odd and at both k=(n+2)/2 and k=(n+4)/2 for n even (Theorem 7 of Alimpiev & Rosenberg 2021).

%C Rows of T(n,k) have a symmetry: T(n,k) = T(n,n-k+3) for all (n,k) with 3 <= k <=n (Theorem 14 of Alimpiev & Rosenberg 2021).

%H Egor Alimpiev and Noah A. Rosenberg, <a href="https://doi.org/10.1016/j.aam.2021.102265">Enumeration of coalescent histories for caterpillar species trees and p-pseudocaterpillar gene trees</a>, Adv. Appl. Math. 131 (2021), 102265.

%F T(n,k) = Sum_{l=k-1..n-1} (l-k+2)^2 * (l-k+4) * (2n-k-l-1)! * (l+k-3)! / ( (l+1)! * (n-l-1)! * (n-k+1)! * (k-3)! ) (Equation 9 of Alimpiev & Rosenberg 2021).

%e Triangle begins (the first row is n=3 and the first column is k=3):

%e     1;

%e     3;     3;

%e     9;    11;     9;

%e    28;    37;    37;    28;

%e    90;   124;   134;   124;    90;

%e   297;   420;   473;   473;   420;   297;

%e  1001;  1441;  1665;  1735;  1665;  1441;  1001;

%e  3432;  5005;  5885;  6291;  6291;  5885;  5005;  3432;

%e 11934; 17576; 20930; 22766; 23354; 22766; 20930; 17576; 11934;

%e 41990; 62322; 74932; 82537; 86149; 86149; 82537; 74932; 62322; 41990;

%t T[n_,k_]:=Sum[(l-k+2)^2*(l-k+4)*(2n-k-l-1)!*(l+k-3)!/((l+1)!*(n-l-1)!*(n-k+1)!*(k-3)!),{l,k-1,n-1}]; Table[T[n,k],{n,3,12},{k,3,n}]//Flatten (* _Stefano Spezia_, Feb 02 2025 *)

%Y T(n,3) and T(n,n) are A000245 (shifted).

%Y T(n,4) and T(n,n-1) are A306423.

%K nonn,tabl

%O 3,2

%A _Noah A Rosenberg_, Feb 01 2025