2,2

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 = 2 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 = 2 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), A145598 (r = 3) 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) = 3/(n+1)*binomial(n+1,k+2)*binomial(n+1,k-1) for n >=2 and 1 <= k <= n-1. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n,2). Row sums A003517.

O.g.f. for column k+2: 3/(k + 1) * y^(k+3)/(1 - y)^(k+5) * Jacobi_P(k,3,1,(1 + y)/(1 - y)).

Identities for row polynomials R_n(x) := sum {k = 1..n-1} T(n,k)*x^k:

x^2*R_(n-1)(x) = 3*(n-1)*(n-2)/((n+1)*(n+2)*(n+3)) * sum {k = 0..n} binomial(n + 3,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 = 6/(n+4)*binomial(2n+1,n-2)*x^n = A003517(n)*x^n.

Row generating polynomial R_(n+2)(x) = 3/(n+3)*x*(1-x)^n * Jacobi_P(n,3,3,(1+x)/(1-x)). [From Peter Bala (pbala(AT)toucansurf.com), Oct 31 2008]

Triangle starts

n\k|..1.....2....3.....4.....5.....6

====================================

.2.|..1

.3.|..3.....3

.4.|..6....15....6

.5.|.10....45...45....10

.6.|.15...105..189...105....15

.7.|.21...210..588...588...210....21

...

Row 4: T(4,1) = 6: the 6 walks of length 4 from (0,0) to (-2,2) are

LLUU, LULU, LUUL, ULLU, ULUL and UULL. Changing L to R in these

walks gives the 6 walks from (0,0) to (2,2).

T(4,2) = 15: the 15 walks of length 4 from (0,0) to (0,2) are

UUUD, UULR, UURL, UUDU,URUL, ULUR, URLU, ULRU, RUUL, LUUR, RLUU,

LRUU, RULU, LURU and UDUU.

.

.

*......*......*......y......*......*......*

.

.

*......6......*.....15......*......6......*

.

.

*......*......*......*......*......*......*

.

.

*......*......*......o......*......*......* x axis

.

with(combinat):

T:= (n, k) -> 3/(n+1)*binomial(n+1, k+2)*binomial(n+1, k-1):

for n from 2 to 11 do

seq(T(n, k), k = 1..n-1);

end do;

A003517 (row sums), A001263, A145596, A145598, A145599, A145601.

Sequence in context: A123289 A096572 A110523 this_sequence A143418 A092370 A006807

Adjacent sequences: A145594 A145595 A145596 this_sequence A145598 A145599 A145600

easy,nonn,tabl

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