3,2

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 generalised 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).

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

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)). [From Peter Bala (pbala(AT)toucansurf.com), Oct 31 2008]

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

.

with(combinat):

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;

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

Sequence in context: A050339 A087288 A095009 this_sequence A117881 A006343 A161433

Adjacent sequences: A145595 A145596 A145597 this_sequence A145599 A145600 A145601

easy,nonn,tabl

Peter Bala (pbala(AT)toucansurf.com), Oct 15 2008