OFFSET
2,1
COMMENTS
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.
LINKS
Stephen B. Connor and Christopher J. Fewster, Integrals of incomplete beta functions, with applications to order statistics, random walks and string enumeration, Brazilian Journal of Probability and Statistics 2022, Vol. 36, No. 1, 185-198; arXiv version, arXiv:2104.12216 [math.CA], 2021.
FORMULA
EXAMPLE
Triangle starts:
2;
16, 6;
90, 52, 20;
448, 306, 180, 70;
2100, 1568, 1086, 644, 252;
...
MAPLE
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)];
MATHEMATICA
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}]
CROSSREFS
KEYWORD
AUTHOR
Christopher J. Fewster, Feb 14 2022
STATUS
approved