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A351585 Triangle read by rows: T(n,k) = A(k,n-k), 1 <= k < n, 2 <= n, where A(m,n) is the number of distinct strings consisting of one X, m+n-1 Y's and m+n-1 Z's in which the X lies to the right of at least m Y's and at least m Z's.

%I A351585 #13 Feb 15 2022 07:21:46
%S A351585 2,16,6,90,52,20,448,306,180,70,2100,1568,1086,644,252,9504,7500,5664,
%T A351585 3948,2352,924,42042,34452,27450,20840,14580,8712,3432,183040,154154,
%U A351585 127380,101950,77640,54450,32604,12870,787644,677248,574574,476652,382510,291896,205062,122980,48620
%N A351585 Triangle read by rows: T(n,k) = A(k,n-k), 1 <= k < n, 2 <= n, where A(m,n) is the number of distinct strings consisting of one X, m+n-1 Y's and m+n-1 Z's in which the X lies to the right of at least m Y's and at least m Z's.
%C A351585 The general string enumeration problem of counting strings with k+k'-1 X's, m+m' Y's and n+n' Z's in which the k'th X is placed after at least m of the Y's and n of the Z's may be expressed in terms of an integral of incomplete Beta functions and evaluated in terms of Kampe de Feriet functions (see Connor & Fewster, 2022). Other special cases include A351583 and A351584.
%H A351585 Stephen B. Connor and Christopher J. Fewster, <a href="https://doi.org/10.1214/21-BJPS522">Integrals of incomplete beta functions, with applications to order statistics, random walks and string enumeration</a>, Brazilian Journal of Probability and Statistics 2022, Vol. 36, No. 1, 185-198; <a href="https://arxiv.org/abs/2104.12216">arXiv version</a>, arXiv:2104.12216 [math.CA], 2021.
%F A351585 T(n+2,1) = A(1,n) = 2*n*binomial(2*n,n-1) = A253487(n-1).
%F A351585 T(m+1,m) = A(m,1) = binomial(2*m,m) = A000984(m) [central binomial coefficients].
%F A351585 T(n,k) = (n - k)*binomial(2*n - 1, n) - 2*k*(n - k)*binomial(2*k - 1, k)*binomial(2*n - 2*k - 1, n - k)/n. See Connor & Fewster (2022).
%e A351585 Triangle starts:
%e A351585      2;
%e A351585     16,    6;
%e A351585     90,   52,   20;
%e A351585    448,  306,  180,  70;
%e A351585   2100, 1568, 1086, 644, 252;
%e A351585   ...
%p A351585 T:=(n,k)->(n - k)*binomial(2*n - 1, n) - 2*k*(n - k)*binomial(2*k - 1, k)*binomial(2*n - 2*k - 1, n - k)/n; [seq(seq(T(n,k),k=1..n-1),n=2..10)];
%t A351585 t[n_,k_]:=(n-k)*Binomial[2*n-1,n]-(2*k*(n-k)/n)*Binomial[2*k-1,k]*Binomial[2*(n-k)-1,n-k]; Table[t[n,k],{n,2,10},{k,1,n-1}]
%Y A351585 Cf. A000984, A253487, A351583, A351584.
%K A351585 nonn,tabl,easy
%O A351585 2,1
%A A351585 _Christopher J. Fewster_, Feb 14 2022