OFFSET
4,2
COMMENTS
T(n,k) is the number of walks of n unit steps, each step in the direction either up (U), down (D), right (R) or left (L), starting from (0,0) and finishing at lattice points on the horizontal line y = 4 and which remain in the upper half-plane y >= 0. An example is given in the Example section below.
The current array is the case r = 4 of the generalized Narayana numbers N_r(n,k) := (r + 1)/(n + 1)*binomial(n + 1,k + r)*binomial(n + 1,k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with an offset of 0 in the row numbering). For other cases see A145596 (r = 1), A145597 (r = 2) and A145598 (r = 3).
LINKS
F. Cai, Q.-H. Hou, Y. Sun, A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 Table 2.1 for k=4.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6
FORMULA
T(n,k) = 5/(n+1)*binomial(n+1,k+4)*binomial(n+1,k-1) for n >=4 and 1 <= k <= n-3. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n + 2,4). Row sums A003519.
O.g.f. for column k+2: 5/(k + 1) * y^(k+5)/(1 - y)^(k+7) * Jacobi_P(k,5,1,(1 + y)/(1 - y)).
Identities for row polynomials R_n(x) := sum {k = 1..n-3} T(n,k)*x^k:
x^4*R_(n-1)(x) = 5*(n - 1)*(n - 2)*(n - 3)*(n - 4)/((n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)) * sum {k = 0..n} binomial(n + 5,k) * binomial(2n - k,n) * (x - 1)^k;
sum {k = 1..n} (-1)^k*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n-2) = A003519(n)*x^(n-2).
Row generating polynomial R_(n+4)(x) = 5/(n+5)*x*(1-x)^n * Jacobi_P(n,5,5,(1+x)/(1-x)). [From Peter Bala, Oct 31 2008]
EXAMPLE
Triangle starts
n\k|...1......2......3......4......5......6
===========================================
.4.|...1
.5.|...5......5
.6.|..15.....35.....15
.7.|..35....140....140.....35
.8.|..70....420....720....420.....70
.9.|.126...1050...2700...2700...1050....126
...
T(5,2) = 5: the 5 walks of length 5 from (0,0) to (1,4) are
UUUUR, UUURU, UURUU, URUUU and RUUUU.
MAPLE
with(combinat):
T:= (n, k) -> 5/(n+1)*binomial(n+1, k+4)*binomial(n+1, k-1):
for n from 4 to 13 do
seq(T(n, k), k = 1..n-3);
end do;
MATHEMATICA
Table[5/(n+1) Binomial[n+1, k+4]Binomial[n+1, k-1], {n, 4, 20}, {k, 0, n}]/.(0-> Nothing)//Flatten (* Harvey P. Dale, Jan 25 2021 *)
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Oct 15 2008
STATUS
approved