A351584 -id:A351584 - cp's OEIS frontend

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, 2m-1 Y's and 2n-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

Christopher J. Fewster, Feb 14 2022

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.

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

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.

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

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)];

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}]

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]

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.

Original entry on oeis.org

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

Christopher J. Fewster, Feb 14 2022

			Triangle starts:
     2;
    16,    6;
    90,   52,   20;
   448,  306,  180,  70;
  2100, 1568, 1086, 644, 252;
  ...

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.

Cf. A000984, A253487, A351583, A351584.

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[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}]

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).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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