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A351584 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, 2*m Y's and 2*n Z's in which the X lies to the right of at least m Y's and at least n Z's.

%I A351584 #15 Feb 15 2022 07:21:18
%S A351584 16,53,53,124,306,124,240,1103,1103,240,412,3043,5664,3043,412,651,
%T A351584 7056,21095,21095,7056,651,968,14476,63480,101950,63480,14476,968,
%U A351584 1374,27114,163986,386249,386249,163986,27114,1374,1880,47331,377616,1226540,1798776,1226540,377616,47331,1880
%N A351584 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, 2*m Y's and 2*n Z's in which the X lies to the right of at least m Y's and at least n Z's.
%C A351584 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 A351585.
%H A351584 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], 201.
%F A351584 T(n,k) = (4*(n - k)*k + 3*n + 2)*binomial(2*n + 2, 2*k + 1)/(4*n + 4) - n*binomial(n, k)^2/2. See Connor & Fewster (2022).
%e A351584 Triangle starts:
%e A351584    16;
%e A351584    53,   53;
%e A351584   124,  306,  124;
%e A351584   240, 1103, 1103,  240;
%e A351584   412, 3043, 5664, 3043, 412;
%e A351584   ...
%p A351584 T:=(n,k)->(4*(n - k)*k + 3*n + 2)*binomial(2*n + 2, 2*k + 1)/(4*n + 4) - n*binomial(n, k)^2/2; [seq(seq(T(n,k),k=1..n-1),n=2..10)];
%t A351584 t[n_,k_]:=(4*k*(n-k)+3*n+2)/(4*n+4)*Binomial[2*n+2,2*k+1]- (n/2)*Binomial[n,k]^2; Table[t[n,k],{n,2,10},{k,1,n-1}]
%Y A351584 Cf. A351583, A351585.
%K A351584 nonn,tabl,easy
%O A351584 2,1
%A A351584 _Christopher J. Fewster_, Feb 14 2022