A351583 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-1 Y's and 2*n-1 Z's in which the X lies to the right of at least m Y's and at least n Z's.

2, 7, 7, 15, 52, 15, 26, 192, 192, 26, 40, 510, 1086, 510, 40, 57, 1115, 4098, 4098, 1115, 57, 77, 2142, 12075, 20840, 12075, 2142, 77, 100, 3752, 30072, 79600, 79600, 30072, 3752, 100, 126, 6132, 66276, 249408, 382510, 249408, 66276, 6132, 126

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 A351584 and A351585.

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+1,1) = A(1,n) = 1/2*n*(3*n+1) = A005449(n), the n-th second pentagonal number.

T(n,k) = 1/(2*Beta(2*k, 2*n - 2*k)) - binomial(n, k)/(2*Beta(k, n - k)), where Beta(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y) is the Beta-function (see A003506). [Connor and Fewster]

Example

Triangle starts:
   2;
   7,   7;
  15,  52,   15;
  26, 192,  192,  26;
  40, 510, 1086, 510, 40;
  ...

Maple

T:=(n, k) -> 1/(2*Beta(2*k, 2*n - 2*k)) - binomial(n, k)/(2*Beta(k, n - k)); [seq(seq(T(n, k), k=1..n-1), n=2..10)];

Mathematica

t[n_, k_]:=1/(2*Beta[2*k, 2*n-2*k])-Binomial[n, k]/(2*Beta[k, n-k]); Table[t[n, k], {n, 2, 10}, {k, 1, n-1}]

Crossrefs

Cf. A003506 (1/Beta), A005449 (column k=1), A351584, A351585.

Sequence in context: A087385 A168278 A267499 * A090521 A090523 A164314

Adjacent sequences: A351580 A351581 A351582 * A351584 A351585 A351586

Keyword

nonn,tabl,easy

Author

Christopher J. Fewster, Feb 14 2022

Status

approved