A145598 Triangular array of generalized Narayana numbers: T(n, k) = 4*binomial(n+1, k+3)*binomial(n+1, k-1)/(n+1).

1, 4, 4, 10, 24, 10, 20, 84, 84, 20, 35, 224, 392, 224, 35, 56, 504, 1344, 1344, 504, 56, 84, 1008, 3780, 5760, 3780, 1008, 84, 120, 1848, 9240, 19800, 19800, 9240, 1848, 120, 165, 3168, 20328, 58080, 81675, 58080, 20328, 3168, 165, 220, 5148, 41184, 151008

Offset: 3

Author

Peter Bala, Oct 15 2008

Keywords

• easy
• nonn
• tabl

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).

Examples

			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
.
		

Links

• F. Cai, Q.-H. Hou, Y. Sun, and 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.

Crossrefs

Cf. A003518 (row sums), A001263, A145602, A145596, A145597, A145599.

Programs

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;

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 2020