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.

2, 16, 6, 90, 52, 20, 448, 306, 180, 70, 2100, 1568, 1086, 644, 252, 9504, 7500, 5664, 3948, 2352, 924, 42042, 34452, 27450, 20840, 14580, 8712, 3432, 183040, 154154, 127380, 101950, 77640, 54450, 32604, 12870, 787644, 677248, 574574, 476652, 382510, 291896, 205062, 122980, 48620

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

Table of n, a(n) for n=2..46.

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

T(n+2,1) = A(1,n) = 2*n*binomial(2*n,n-1) = A253487(n-1).

T(m+1,m) = A(m,1) = binomial(2*m,m) = A000984(m) [central binomial coefficients].

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).

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

Cf. A000984, A253487, A351583, A351584.

Sequence in context: A087251 A336833 A211367 * A220314 A302206 A110008

Adjacent sequences: A351582 A351583 A351584 * A351586 A351587 A351588

Keyword

nonn,tabl,easy

Author

Christopher J. Fewster, Feb 14 2022

Status

approved