OFFSET
3,2
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 = 3 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 = 3 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 A145599 (r = 4).
LINKS
F. Cai, Q.-H. Hou, Y. Sun, A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018; Table 2.1 for k=3.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
FORMULA
T(n,k) = 4/(n+1)*binomial(n+1,k+3)*binomial(n+1,k-1) for n >=3 and 1 <= k <= n-2. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n + 1,3). Row sums A003518.
O.g.f. for column k+2: 4/(k + 1) * y^(k+4)/(1 - y)^(k+6) * Jacobi_P(k,4,1,(1 + y)/(1 - y)).
Identities for row polynomials R_n(x) = Sum_{k = 1 .. n - 2} T(n,k)*x^k:
x^3*R_(n-1)(x) = 4*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)*(n + 4)) * Sum_{k = 0..n} binomial(n + 4,k) * binomial(2n - k,n) * (x - 1)^k;
Sum_{k = 1..n} (-1)^(k+1)*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n-1) = A003518(n)*x^(n-1).
Row generating polynomial R_(n+3)(x) = 4/(n+4)*x*(1-x)^n * Jacobi_P(n,4,4,(1+x)/(1-x)). - Peter Bala, Oct 31 2008
G.f.: A(x)=x*A145596(x)^2. - Vladimir Kruchinin, Oct 09 20
EXAMPLE
Triangle starts
n\k|..1.....2....3.....4.....5.....6
====================================
.3.|..1
.4.|..4.....4
.5.|.10....24...10
.6.|.20....84...84....20
.7.|.35...224..392...224....35
.8.|.56...504.1344..1344...504....56
...
Row 5:
T(5,3) = 10: the 10 walks of length 5 from (0,0) to (2,3) are
UUURR, UURUR, UURRU, URUUR, URURU, URRUU, RUUUR, RUURU, RURUU
and RRUUU.
*
*......*......*......y......*......*......*
.
.
*.....10......*.....24......*.....10......*
.
.
*......*......*......*......*......*......*
.
.
*......*......*......*......*......*......*
.
.
*......*......*......o......*......*......* x axis
.
MAPLE
T := (n, k) -> 4/(n+1)*binomial(n+1, k+3)*binomial(n+1, k-1):
for n from 3 to 12 do seq(T(n, k), k = 1 .. n-2) end do;
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Oct 15 2008
STATUS
approved