login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A145596 Triangular array of generalized Narayana numbers: T(n,k) = 2/(n+1)*binomial(n+1,k+1)*binomial(n+1,k-1). 8
1, 2, 2, 3, 8, 3, 4, 20, 20, 4, 5, 40, 75, 40, 5, 6, 70, 210, 210, 70, 6, 7, 112, 490, 784, 490, 112, 7, 8, 168, 1008, 2352, 2352, 1008, 168, 8, 9, 240, 1890, 6048, 8820, 6048, 1890, 240, 9, 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

T(n,k) is the number of walks of n unit steps on the square lattice (i.e., each step in the direction either up (U), down (D), right (R) or left (L)) starting from (0,0) and finishing at points on the horizontal line y = 1, 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 = 1 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 row numbering starting at n = 0). For other cases see A145597 (r = 2), A145598 (r = 3) and A145599 (r = 4).

LINKS

Table of n, a(n) for n=1..55.

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

T. Mansour, Y. Sun, Identities involving Narayana polynomials and Catalan numbers

FORMULA

T(n,k) = 2/(n + 1)*binomial(n + 1,k + 1)*binomial(n + 1,k - 1) for 1 <= k <= n. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n - 1,1).

O.g.f. for column (k + 2): 2/(k + 1) * y^(k+2)/(1 - y)^(k+4) * Jacobi_P(k,2,1,(1 + y)/(1 - y)). The column generating functions begin: column 2: 2*y^2/(1 - y)^4; column 3: y^3*(3 + 2*y)/(1 - y)^6; column 4: y^4*(4 + 8*y + 2*y^2)/(1 - y)^8; the polynomials in the numerators are the row generating polynomials of array A108838.

O.g.f. for array: 1/(2*x*y^3) * {((1 + x)*y - 1)*sqrt[1 - 2*(1 + x)*y + (y - x*y)^2] + x^2*y^2 - 2*x*y + (1 - y)^2} = x*y + (2*x + 2*x^2)*y^2 + (3*x + 8*x^2 + 3*x^3)*y^3 + (4*x + 20*x^2 + 20*x^3 + 4*x^4)*y^4 + ....

Row sums A002057.

Identities for row polynomials R_n(x) := sum {k = 1..n} T(n,k)*x^k (compare with the results in section 1 of [Mansour & Sun]):

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

R_n(x) = sum {k = 0..floor((n-1)/2)} binomial(n,2*k + 1) * Catalan(k + 1) * x^(k+1)*(1 + x)^(n-2k-1);

Sum {k = 1..n} (-1)^(n-k)*binomial(n,k)*R_k(x)*(1 + x)^(n-k) = x^m*Catalan(m) if n = 2*m - 1 is odd, otherwise the sum is zero.

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) = 4/(n + 3)*binomial(2*n + 1,n - 1)*x^(n+1) = A002057(n-1)*x^(n+1).

Row generating polynomial R_(n+1)(x) = 2/(n + 2)*x*(1 - x)^n * Jacobi_P(n,2,2,(1 + x)/(1 - x)). - Peter Bala, Oct 31 2008

EXAMPLE

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

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

.1.|..1

.2.|..2.....2

.3.|..3.....8....3

.4.|..4....20...20.....4

.5.|..5....40...75....40.....5

.6.|..6....70..210...210....70.....6

...

Row 3 entries:

T(3,1) = 3: the 3 walks from (0,0) to (-2,1) of three steps are

LLU, LUL and ULL.

T(3,2) = 8: the 8 walks from (0,0) to (0,1) of three steps are

UDU, UUD, ULR, URL, RLU, LRU, RUL and LUR.

T(3,3) = 3: the 3 walks from (0,0) to (2,1) of three steps are

RRU, RUR and URR.

.

.

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

.

.

*......3......*......8......*......3......*

.

.

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

.

.

MAPLE

with(combinat):

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

for n from 1 to 10 do

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

end do;

CROSSREFS

A002057 (row sums), A001263, A108838, A145597, A145598, A145599, A145600.

Sequence in context: A296952 A141611 A234357 * A186753 A135835 A177696

Adjacent sequences:  A145593 A145594 A145595 * A145597 A145598 A145599

KEYWORD

easy,nonn,tabl

AUTHOR

Peter Bala, Oct 14 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 23 00:33 EST 2018. Contains 299473 sequences. (Running on oeis4.)